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A1110U 3 NBSIR 83-2665

An Investigation of the Forced

Ventilation in Containership Holds

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, DC 20234

May 1983

Sponsored by:

U.S. Coast Guard

Washington, DC

[

i

NATIONAL EUREAU OF STANDARDS

LIBRARY MAY 2 3 1983

OcKoXX. Cxfc

NBSIR 83-2665

AN INVESTIGATION OF THE FORCED

VENTILATION IN CONTAINERSHIP HOLDS

Howard R. Baum

John A. Rockett

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Fire Research

Washington, DC 20234

May 1983

Sponsored by:

U.S. Coast Guard

Washington, DC

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Table of Contents

Page

LIST OF FIGURES Lv

LIST OF TABLES vLL

NOMENCLATURE.... vlll

Abstract 1

1. INTRODUCTION 1

2. BASIC ASSUMPTIONS OF THE THEORETICAL MODEL 10

3. SLOT FLOW IN A STABLY STRATIFIED ENVIRONMENT 12

4. THE SLOT PRESSURE DISTRIBUTION 20

5. THE END VOID PRESSURE 26

6. THE BOTTOM MOTION : 32

7. THE VAPOR PICKUP 42

8. NUMERICAL PROGRAMS 50

8.1 Main Numerical Program..... 50

8.2 Subroutines 54

8.3 Input Data 61

9. NUMERICAL RESULTS 62

10. EXPERIMENTAL PROGRAM 67

11. CONCLUSIONS 71

12. REFERENCES 72

APPENDIX 1. Example of Input Data 107

APPENDIX 2. Output for the Input Data of Appendix 1

APPENDIX 3. Computer Code Listing 120

-ill-
List of Figures

Page

FLgure 1. Cross sectLon of a typLcal containersh Lp near its

mid sectLon. To the right is shown the contaLner

stacking and to the left the bulkhead framing

Figure 2. Vapor pressure versus temperature for heptane.

Rich and lean limits are for an aLr environment

Figure 3.

7 f) Idealized containershlp hold loaded with containers....

Figure 4. Geometry of the space between two stacks of containers

(inter-container slot)

FLgure 5. Conformal mappings used to obtaLn the Greens function,

G, in equation 24 78

FLgure 6. Plan vLew to the end void showing the locatLon of the

forced vent Llat Lon extraction poLnt (suction).

Ventilation flow enters from the top of the void and

from the intercontainer slots (one of N slots indicated

by a vertical line) 79

Figure 7. Cross section of the bottom of an intercontainer slot.

Velocity distribution of the gas as Lt approaches the

slot bottom is indicated next to each wall 80

FLgure 8. Universal velocity profiles for the axial flow in the

boundary layer of a slot bottom. The flow is

symmetrical about the slot centerline so only one half

the slot width is shown. The velocity is zero at the

slot bottom and increases toward an asymptotic profile

as height, y, above the bottom increases... 81

Figure 9. Universal velocity profiles for the transverse flow in

the boundary layer of a slot bottom. The flow is antisymmetric about the slot centerline so only one half

the slot width is shown. The velocity is zero at the

slot bottom and increases toward an asymptotic profile

as height, y, above the slot bottom increases

FLgure 10. Perspective sketch of an intercontainer slot showing

air flow streamlines. The air picks up evaporated

spill vapor as it moves Ln the boundary layer along

the bottom of the slot 83

FLgure 11. Simplified flow diagram for the computer program 84

Figure 12. Universal slot bottom axial velocity plotted versus height for a fixed transverse location (centerline).... 85

-iv-
Pifie

Figure 13. Slot bottom stream functLon near the Intersection

of the slot with the hold end void as a function of

distance from the intersection (at x = -1/2)

Figure 14. Dimensionless slot pressure as a function of

dimensionless height above the slot bottom for two

slots: solid line, slot nearest the suction and

dotted line, slot furthest from the suction.

Pressures calculated single precision. Compare with

figure 20-a which is the same case calculated in

double-precision

Figure 15. Computed mass pick-up by evaporation and removal by

forced ventilation as a function of suction height

above the slot bottom for two different amounts of

stable stratification

Figure 16. Dimensionless mass pick-up as a function of slot

number for a hold containing 11 slots (10 stacks of

containers) for two forced ventilation suction

locations: solid line, suction on hold centerline;

dotted line, suction at the out-board corner of the

hold

Figure 17. Dimensionless pressure as a function of dimensionless

height (double precision calculation) stable stratification 0.1°C over height of hold

86

87

88

89

90

Figure 17a* Physical height of suction 0.05 m, dimensionless

height 0.025

Figure 17b. Physical height of suction 0.60 m, dimensionless

height 0.294

Figure 18. Slot bottom boundary layer streamlines corresponding

to the pressures shown in figure 17a

Figure 18a. Streamlines for slot at center (nearest suction) 92

Figure 18b. Streamlines for slot at hold side (furthest from

suction) 93

Figure 19. Slot bottom boundary layer steamlines corresponding

to the pressures shown in figure 17b. 94

Figure 19a. Streamlines for slot at centerline (nearest suction)... 94

Figure 19b. Streamlines for slot at hold side (furthest from

suction) 95

Figure 20a. Dimensionless pressure as a function of dimensionless

height (double precision calculation, compare with

figure 14). Suction 1 m above hold bottom 96

-v-
Page

Figure 20b. Slot bottom boundary layer streamlines correspond lng

to the pressure shown in figure 20a. Centerline

slot (nearest suet Lon)

Figure 20c. Slot bottom boundary layer streamlines corresponding

to the pressures shown in figure 20a. Side slot

(furthest from suction)

Figure 21a. Dimensionless slot pressure as a function of

dimensionless height (double precision-calculation)

stable stratification 1°C over height of hold

(compare with figure 17)...

Figure 21b. Slot bottom boundary layer streamlines correspond ing

to the pressures shown in figure 21a. Centerline

slot (nearest suction)

Figure 21c. Slot bottom boundary layer streamlines corresponding

to the pressures shown in figure 21a. Side slot

(furthest from suction)

Figure 22. Simplified schematic of experimental apparatus....

Figure 23. Concentration of combustible vapor in the exhaust

duct for several ventilation rates and two suction

locat ions

Figure 24. Concentration to combustible vapor inside the

experimental enclosures versus height of the sampling

point. One air change per hour forced ventilation

flow, low suction

97

98

99

100

101

102

103

104

-vi-
List of Tables

Page

Table 1. Vertical pressure gradient effectiveness f(w)

dependence 105

Table 2. Dimensionless mass pick-up, suction height 5 cm mass pick-up 106

-vii-
Nomenclature

Airy function

Vapor concentration In air

Equilibrium vapor concentration of spill material

Fuel diffusivity in air

Slot half width

Function defined by eq. (10)

Grashof Number

Acceleration of gravity

Slot height

Unit vector in the x direction

Unit vector in the y direction

Unit vector in the z direction

Length of slot "n"

Number of air changes per hour

Local mass flux picked up, defined in eq. (73)

Total mass flux picked up in a slot, defined in eq. (76)

Number of slots

Slot number

Outward pointing normal to integration centure

Thermodynamic pressure

Perturbation due to flow pressure

Prandtl Number

Slot source flow per unit area

Total flow through end void

Total flow through the nth slot

Total flow drawn through hold

-viii-
c

+

q

Re

Sc

T

t

u

v

V

X

y

V

z

Z

V Y

v

6

5

n

C

p

X

u

v

T

Normalized slot flow

2 Reynolds Number = md /vt

Schmidt Number

Temperature

time

Gas velocity

Gas velocity component parallel to x

Gas velocity component parallel to y

Gas velocity component parallel to z

Coordinate parallel to slot or void length

Coordinate parallel to slot or void height

Dimensionless coordinate defined in eq. (38)

Coordinate parallel to slot or void width

Dimensionless coordinate defined in eq. (38)

Location of air extractor in end void

Dirac Delta Function

Dimensionless coordinate parallel to x

Also, real part of complex variable defined in eq. (25)

Dimensionless coordinate parallel to y

Also, imaginary part of complex variable defined in eq. (25)

Dimensionless coordinate parallel to z

Gas density

Concentration layer vertical coordinate defined in eq. (66)

Gas viscosity

Gas kinematic viscosity

Time unit (hours)

Also, complex varible defined in eq. (33)

-ix-
G

U)

r

*

Dimens Lonless temperature

Scale factor defined in eq. (6)

Gamma function

Stream function defined in eq. (64)

Subscripts

0 Reference value for T, p

1 Value at top of hold

n Value associated with nth slot

v Value associated with end void

Superscripts

* Dimensionless, perturbation values, defined in eq. (38)

Dimensionless, perturbation value, defined in section 3

~ Dimensionless variables defined In eq. (20)

Operators

V Vector gradient

A Laplacian

-x-
AN INVESTIGATION OF THE FORCED VENTILATION IN CONTAINERSHIP HOLDS

Howard R. Baum

John A. Rockett

Abstract

An analysis of the fluid flow and mass transfer Induced by vent Hat Lon

systems in contatnership holds was carried out. The result of the work was

used to support the U.S. position before a committee of the International

Convention on Safety to Life at Sea. The analysis consists of a detailed

calculation of the forced motion through an interconnected set of narrow,

stably stratified vertical air passages, which represent an ideal Lzed

containership hold. The results of this calculation were then used to predLct

the vapor concentration of spilled volatile material assumed to lie at the

bottom of the vertical air passages. The result Ls a set of formulae which

determine the rate of extraction of volatile material as a function of hold

geometry, ventilation parameters, and ambient stratification. The results are

incorporated in a computer program which is described in detail. A variety of

computed results are presented. The results indicate the crucial Importance

of locating the extractor as close to the hold bottom as technically possible.

1. INTRODUCTION

The purpose of this study was to obtain the information necessary to

prepare a quantitative statement on the degree of fire hazard that might exist

in the hold of a large containership as a function of the amount and nature of

the hold ventilation and amount and kind of leakage of flammable liquid or gas

cargo. The effort was mainly analytic although some scale model tests were

- 1 -
conducted. Sea tests were conducted by Sealand Corporation to determine the

degree to which the thermal conditions assumed by the analysis were found In

pract Ice.

Conta Inershlps play a major role in the U.S. Merchant Marine. The

majority of the non-bulk sea-going cargo into and out of the U.S. is now moved

in containers. Some of this cargo is classified as "dangerous cargo", such as

flammable liquids and gases and are regulated in treaty provisions based on

recommendations of the International Maritime Consultative Organization

( IMCO)

Under existing regulations stowage limitations, resulting from the types

and amounts of cargo classed as dangerous, reduce shipment scheduling

flexibility. IMCO has considered new regulations which would somewhat relax

the dangerous category definitions for ships with adequate hold ventilation or

detection and inerting systems. The basis for their action is the belief, by

some members of the committee, that ventilation would keep the concentration

of leaking and vaporizing flammable gas below the flammable or explosive

limit. However, prior to this study, there did not appear to be a rational

basis for establishing a suitable rate of ventilation. A study on the scale

of this one could not address the full range of situations which might arLse

but the most prevalent situation was considered. Using the results of this

study a position was developed and presented by the U.S. representative to

IMCO as a counter to the ad hoc opinions originally put forward by others.

After discussion, a position close to the U.S. position was adopted by the

committee.

-2-
Container freight forwardLng Ls a hLghly automated process using ships

often specifically built for container freight and complex dock side equipment

matched to the ship design. Containers of a uniform size are closely stacked

within the holds, filling all the space available after allowance for

structure, container guides and spaces at the sides, especially near the ends

of the ship, due to the non-rectangular shape of holds. Current practice

loads a hold to its maximum capacity. In the center portion of a ship, with

wing tanks occupying the space between the hull plating and the side combing

of the hatch, the only significant voids would be between the top of the

containers and the underside of the hatch covers; and at one end of the

container stacks, where deep framing of the transverse bulkhead result in a

array of rectangular spaces interconnected by lightening holes in the

frames. The spaces between stacks of containers, between the containers and

the wing tank walls, or between the end of the container stacks and the smooth

(non-framed) side of a transverse bulkhead would be only that needed for the

container guides, typically about 10 cm between stacks of containers and half

this between the containers and bulkhead, see figure 1. On some ships there

may be fixed, heavy longitudinal beams to support the hatch cover. The gaps

between container stacks on either side of such a beam would be somewhat

wider. Where wing tanks are not used and at the ends of the ship, a fairly

large void will exist between the hull plating and the outermost container

stack. This will be partially subdivided by the transverse and longitudinal

framing. The container stacks rest on the flat, smooth double bottom tank

top. Although an air space some 10 cm deep exists under the containers this

is cut off from the inter-container gaps and end void by side and end rails or

skirts of the container so that effective communication with this space Ls

limited by the thickness of the corner pads of the containers, a gap perhaps

- 3-
less than 2 cm. Should a liquid be spilled on the tank top It would be able

to flow under the containers as the ship rolled but liquid evaporated under

the containers would not easily escape to the rest of the hold.

The accident scenario envisaged involves a container carrying general

cargo including some flammable liquid Ln cans or drums. For any of a number

of reasons — a defective drum, inadequate dunnage and securing, rough

handling of the container, etc. — flammable liquid is assumed to escape Into

the interior of the container. Although a container in good condition is

quite weather tight, an older container, especially if it has been roughly

treated (a situation likely to accompany disruption of its cargo), may allow a

liquid, spilled inside, to leak out. It is assumed that liquid does escape

and flows down over the outside of the containers below to the bottom of the

hold, where It accumulates in a puddle which is spread by rolling and pitching

of the ship to wet the entire bottom of the compartment. In the case

considered for the numerical examples used throughout this paper, the liquid

o is heptane and the tank top area is 324 m . If 208 liters (55 U.S. gallons)

of liquid reached the bottom of the hold and spread uniformly over the tank

top, the liquid layer would be only 2/3 mm thick.

Consider the Implications of this accident first as it affects conditions

inside the container and next as It affects conditions In the hold. Inside

the container the air will be essentially stagnant for any plausible hold

ventilation scheme. Accumulated liquid will evaporate reaching a local

equilibrium concentration depending on the container temperature. Hold

temperatures measured by Sealand on a run from Houston to Rotterdam [1]* In

1 Numbers in brackets refer to references listed at the end of this report.

-4 -
the fall of 1978 ranged from 10 to 28°C (50 to 83°F). From figure 2 [2] It is

seen that the equilibrium vapor pressure of heptane, for these temperatures,

ranges from 20 to 53 mm-Hg yielding corresponding volume concentrations of 2.6

to 7.0%. Lewis and Von Elbe [3] give the flammability limits for heptane in

air as 1.2 and 6.7%. Thus the undiluted heptane vapors will lie within the

flammable limits throughout the expected temperature range. Heptane (CyH^)

vapors are heavier than air. The mean molecular weight at standard temperature and pressures of the equilibrium heptane-air mixture for this range of

vapor pressures is 30.84 to 34.11 compared to 28.97 for air. This will tend

to inhibit mixing of the heavy vapors with the rest of this air in the

container but, over a long period of time, through diffusion, a substantial

volume of combustible vapor could accumulate inside the container (121 to

312 gm of heptane/m of air in a container with total volume about 90 m and

void volume estimated at 10 m ). Our accident scenario supposed that the

container would allow the spilled liquid to leak out but, of course, it might

not, or might leak very slowly. Thus, if an ignition source of sufficient

strength to ignite the vapors were found within the container, a vapor

deflagration could occur followed by fire. Recall that this situation is

independent of the amount and type of hold ventilation and that, for the

chemical and temperatures chosen, there is little possibility of escaping

danger by exceeding the rich flammable limit after a long time.

If the container leaks, some or most of the liquid can escape, possibly

alleviating the hazard just described but creating another in the ship's

hold. As already noted the liquid will form a thin but extensive puddle on

the tank top. If there is a low point sump in the hold, a substantial amount

of the spill may drain into it and could be pumped to a safe holding tank.

- 5-
Such an arrangement seems the most suitable way to remove any substantial

amount of liquid. However, evaporation will occur In the hold just as Lt does

In the container, but. In the hold, ventilation can greatly reduce the hazard.

Consider first the zero ventilation situation In an unstratlfled hold.

While, as just discussed, there might be 1 to 3 kilograms of fuel vapor in the

container where the spill originated, the ship's hold is so large that, for a

plausible spill volume, evaporation followed by thorough mixing might exhaust

the available vapor before the lean limit concentrations were reached. For a

hypothetical ship with wing tanks, i.e. , minimal side voids* loaded with 10

rows of 35 foot containers across the hold each row 7 containers high, the

o o total hold volume might be about 6,500 m (230,000 ft ) and the void volume

1,760 m (62,000 ft ). This void volume could carry, for the range of temperatures expected, from 210 to 550 kg heptane, or approximately the contents of

1-1/3 to 3-1/2 55 gallon drums. A very large ship without wing tanks would

have considerably more void volume in its midships holds while, in the ends a

ship, there could be considerably less void volume. In any event, the advantage of a low point sump becomes clearer in this context since it could remove

much of the spill liquid before it evaporated. As noted earlier, the film of

liquid from a 55 gallon spill, spread evenly over the tank top of a large ship

Is quite thin. Since some residual film, puddles in irregular low spots,

etc., must be anticipated, there would be a significant reservoir for evaporation even with an efficient sump. This residual volume is difficult to

estimate but, for a low viscosity liquid might be about 20 to 30 gallons In so

o large a space. For the void volume of our example (1760 m ) at the lean limit

concentration there would be 94 kilograms of heptane vapor or the result of

evaporating 33 gallons. Thus, by using a sump, there is the possibility of

keeping the average composition of the vapor below the flammable limit.

- 6 -
Although the average composition of the vapor might be kept below the

flammable limits, during the evaporation process (while liquid remains) there

will be a region near the liquid surface in which the vapor concentration will

approach the equilibrium vapor concentration corresponding to the liquid

temperature. As we have seen, this can be expected to be well within the

flammable limits. The purpose of ventilation is to keep the volume of gas

which is within the flammable limits as small as possible. If the air in the

hold were continually stirred, for example, by natural convection created by

an unstable vertical temperature gradient — bottom of the hold warmer than

the top, the same situation described under zero ventilation would apply.

However, an unstable temperature gradient was observed only intermittently on

the instrumented containership run from Houston to Rotterdam and then only in

the upper portion of the hold. The lowest thermocouple was always the

coolest. Throughout most of the voyage the hold air was stably stratified for

all heights measured. In addition to the temperature stratification, if there

are pools of flammable liquid at the bottom of the hold, the vapor just above

these pools will be heavier than pure air. This may be expressed as an added

equivalent thermal stratification by giving the temperature difference

required to produce the same density difference in pure air as is produced by

the fuel vapor. For heptane at the temperature observed, this ranges from

about 20 to 50°C. By contrast the true thermal stratification on the

instrumented sea run never exceeded 3°C, and was more typically less than

1°C. Thus the combination of a stable temperature field and heavy evaporated

liquid vapor tends to be extremely stable near the tank top (hold bottom) and

generally stable, though much less so, elsewhere.

- 7 -
If there were a transverse temperature difference, one side of the hold

warmer than the other, a circulation would develop [A, 5]. In an empty hold a

narrow boundary layer flow would move up the warmer side across the top of the

hold and down the cooler side. Near-stagnant conditions would be found in the

interior of the hold. The flow across the tank top would also be confined to

a thin boundary layer. In a loaded ship, due to the presence of the container

stacks, this flow would be strongly inhibited except in the end void associated with the bulkhead framing and here the framing would considerably reduce

the general flow. In practice, the flow induced by a transverse temperature

gradient in the presence of a stable vertical gradient, would probably only be

significant in the two side voids of a ship without wing tanks. The circulation would be between the sides of the ship and the outer side of the

outermost container stacks. Such flow has not been considered in this study.

The flow that seems most likely to affect the vapor bubble over

evaporating liquid on the tank top is that associated with forced vent nation. Obviously, for the well mixed case (unstable stratification) the

location of the suction and inlet for the forced flow are relatively unimportant although they should be well separated. In the stably stratified case

this is not true. Both since the stable case is more prevalent and since, in

the unstable case, the suction may be located anywhere and might as well be

placed advantageously for the stable situation, the stable case has been given

priority in our study. With stable stratification and flammable vapors

heavier than air originating from a liquid spill, the suction should be close

to the bottom of the hold and the inlet placed well above it. As will be

discussed in detail in the following sections, the air flow will at first

spread laterally from the inlet with very limited vertical movement. There

- 8-
will be a relatively slow drLft downward to the level of the suction followed

by lateral movement In the plane of the suction, again with little vertical

motion, to the suction location. If the suction is located above the tank top

(at the bottom of the hold shown in figure 1) the gas below the suction will

tend to be stagnant. In the stably stratified case, vertical movement of the

gas Is facilitated where it can exchange heat with its surroundings. The

result is that the vertical drift from the level of the inlet to that of the

suction is not uniform but concentrated in thin boundary layers adjacent to

the container stacks and ship structure. The more stable the stratification

the narrower these boundary layers become. For the geometry and temperature

differences found in a typical container ship these boundary layers are only a

few centimeters thick. The result is that virtually the same flow can move

down the 10 cm wide gap between container stacks as down the several meter

wide gap between the outermost container stack and the side of the ship. Only

when the gap is narrower than the combined thickness of the two boundary

layers is the flow decreased. This may occur in the gap between the end of

the container stacks and the smooth side of the bulkhead.

In all the above, the accident was assumed to involve a liquid spill.

Although this appears to be the most likely type of accident, some materials

could be released whose vapors are lighter than air. To deal with this

eventuality, it has been proposed that the forced ventilation inlet be located

near but not at the top of the hold and that a suction pulling a minor

fraction of the ventilation be provided at the highest point in the hold, just

under the hatch cover.

-9 -
2. BASIC ASSUMPTIONS OF THE THEORETICAL MODEL

In order to develop a quant LtatLve model, it Ls necessary to know the

geometry and thermal strat If Icat Lon of a typical container ship hold. The

most obvious feature of such holds (on efficiently designed shLps) Ls that

most of the available space is occupied by containers. The only aLr spaces

are narrow vertical slots between stacks of containers, similar but less

narrow voids at ends and/or sides of the stacks, and a gap between the top of

the container stacks and the hatches. The size and shape of these vary from

ship to ship, and from one hold to the next on a gLven ship. The temperature

distrLbutLon in each hold is dependent on both the ship and its thermal

environment over a period of time. In general, the environment Ls highly

dependent on the shLps route and both seasonal and dally weather patterns.

The conditions prevailing Ln tanks adjacent to the hold are also important and

may vary markedly during a voyage. Given this environment, the complete

determination of the thermal balance on a ship is itself a formidable task.

Rather than attempt to model the detailed features of a single hold and

thermal environment, a set of simplifying assumptions is introduced, which

permits the analysis to be reduced to a tractable size and scope, and still

retain some dependence on the physical and geometric parameters described

above. These assumptions are:

(1) The hold is rectangular. The air spaces consist of narrow

rectangular vertical slots separating container stacks and a

narrow rectangular vertical void at one end of the hold. The

Ldealized hold ls shown schematically in figure 3.

- 10-
(2) The temperature distribution in the hold is stably stratified

and varies linearly from top to bottom. The containers and

ship hull are in thermal equilibrium with this distribution.

Thus, all motions are due to ventilation.

(3) The ventilation system is designed so that air enters at the

top of the hold and exits in the end void. The overall air

volume flow is consistent with creeping motion (inertia forces

unimportant)

Finally, in order to estimate the rate at which spilled material is

picked up it is necessary to impose a spill scenario on the model. It is

assumed that the spill material collects at the bottom of the slots between

container stacks. The material is picked up as vapor in a concentration

boundary layer formed at the bottom of the slot. All material caught up in

this boundary layer is assumed to exit with the ventilation air. The analysis

then proceeds as follows:

First, the conditions for low Reynolds number flow are established and

the small scale motion in a single slot is determined. This leads to an

equation for the pressure that governs the large scale motion in a single

slot. This equation is then solved assuming that the pressure in the end void

where the flow exits is known. The next step is the solution for the pressure

in the end void, which ties together the large scale motion in the entire

hold. Then the local flow in the bottom of each slot is obtained. The final

step is the calculation of the concentration boundary layer in the slot

bottom, which determines the actual pickup of spill material.

- 11-
3. SLOT FLOW IN A STABLY STRATIFILD ENVIRONMENT

As mentioned above, the volume available for air movement In a containership hold may be usefully idealized as a collection of narrow vertical and

horizontal slots. The analysis of the motion in a single slot Ls thus a

necessary precondition for a study of the air movement throughout the hold.

In order to proceed, we must first establish that the creeping flow regime Ls

encountered for realLstLc values of the governing flow parameters. Then,

approximate solutions to the equations of motion valid in the appropriate flow

regime can be constructed. Finally, these solutions will be related to the

large scale motion in the hold.

ConsLder a vertically oriented slot of width 2d, height h and length Jl

(fLgure 4). The equations governing the steady motion of a viscous incompressible fluid affected by buoyancy forces can be written in the BoussLnesq

approximation as:

•> V • u = o

(u V)

-* 1 * u + — Vp +

P 8 - vAu ( 1 )

(u • V) T = ( v/Pr)AT

Here u is the velocity vector, p the pressure, and T the temperature in the

fluid. The densLty p, kinematLc viscosity v and Prandtl number Pr are

properties of the fluid taken as constant corresponding to the hold bottom

- 12-
temperature T

Q

. The gravitational acceleration g Is directed vertically

downward, while V and A are respectively the gradient and Laplaclan operators.

Equation (1) represents the conservation of mass, momentum, and energy

respectively. It is anticipated that the ventilation system will be designed

to induce mean (mass averaged) velocities in the slot whose order of magnitude

is such that air actually in the hold bottom can be swept out several times

per hour. Let m be the number of times per hour that a slot is swept out

horizontally. Then a typical horizontal velocity must be of order m£/T,

where t is the period (one hour). The inertial terms (the non-linear terms)

2 in the horizontal momentum balance are then of order (m£/x) /£; while the

2 viscous terms are of order v(m£/x)/d . The ratio of these two terms indicates

the relative importance of viscous and inertial effects. It will be called

the "effective Reynolds number" in this report to distinguish it from more

conventionally defined Reynolds numbers. This effective Reynolds number (Re)

for horizontal motion in the slot, which determines the flow regime of

interest, is given by

For slot widths and sweep rates of interest the effective Reynolds number Re

is typically in the range 1 _< Re 10.

For this range of values, the horizontal flow is effectively one in which

the pressure forces balance the viscous forces on the fluid, as in pipe flow

and in bearing lubrication [6,7], This occurs because the slot is so narrow

in comparison to Its length (d/Jl << 1) that velocities perpendicular to the

- 13 -
plane of the slot (L.e. - In the z-direclton, see figure 4) are negligibly

small compared with those in the plane of the slot. Although the above

argument strictly applies to horizontal motions, it will be shown in detail

below that the effect of stable stratification will be to reduce even further

the importance of inertial effects on the fluid motion.

We now turn to a detailed study of the motion in the slot. Let P

Q

and T

o be reference values of the pressure and temperature of the air in the slot.

The fluid velocity u may be expressed in component form (see figure 4) as:

u=ui+vj+wk.

The dependent variables describing the state of motion may be represented

as follows:

P = P

Q " Pgy + P *

T = T

q + (T

x - T

q

) y/h + 0(x/A, y/h, z/d)

u = u(x/Z, y/h, z/d) ( 2 )

v = v(x/£, y/h, z/d)

w = w (x

/

SL

y y/h, z/d)

- 14-
Here, g is the magnitude of the gravitational acceleration and p Is the

dynamical part of the pressure. The remaining terms in the expression for p

are the hydrostatLc values of the pressure. The term linear In (y/h) In the

expression for the temperature Is the ambient stratification of the hold.

This stratification is assumed to vary linearly between the upper temperature

Tj and Tq, where T^ > Tq. The velocity component normal to the plane of the

slot, w, is smaller than the in-plane components u and v by a factor d/Jl or

d/h. The conservation of momentum in the z direction immediately leads to the

conclusion that the dynamic pressure p must be nearly independent of z.

It is convenient to work with non-dimensional variables defined as

follows

p = (Re)pgh [ -J p(S, n)

v = (Re)gd -j

v(£, n, O

6 = (Re)(T

1 - To )0(5, n, C)

(3)

K * X n , n = y/h, c = z/d

Substitution of the non-d Lmensional variables defined in equation (3) into the

vertical momentum and energy conservation equations and neglecting terms of

order (d/h)^, (d/Jl)^, or Re yields:

- 15-
V

( G)~ 1 /2

Pr

(h/d) ( 4 )

The dimensionless parameter G, the Grashof number, is the fundamental

parameter controlling the nature of the vertical motion in the slot. Its

Influence will be discussed In detaLl below.

Since p(C,n) is independent of £, equation (4) can be solved for the

dimensionless vert Leal velocity v and temperature perturbation 0 as functions

of the vertical pressure gradient -r^-. The boundary conditions associated with

dp

equation (4) are:

v(— 1 ) = v(l) = 0

0(-l) = 0(1) = 0

(5)

The physical meaning of equation (5) is that the vert Leal velocity and

temperature perturbation must vanish at the sides of the slot. The first

boundary condition follows from the no-slip condition. The second comes from

the assumption that the ambient stratification in the slot Ls controlled by

the temperature distribution in the containers, which varies linearly with

height.

The solution to equations (4) and (5) is given by:

- 16 -
aO) = -

sin(a)) slnh(u))

( 6 ) 2 2 -sin ( a> ) + cosh (w)

b(w) = -

cosh(oj) cos(to)

2 2 -sin (w) + cosh (a)) — (GPrd/h) 1/4

/2

It should be noted that the above solution, while approximate in terms of the

overall problem of Interest, is in fact an exact solution of the equatLons of

hydrodynamics for an infinitely long slot. This buoyancy layer was first

found by Prandtl [8], and by Gill [9], and used by Gill [9], in his analysis

of thermally driven slot convection. Gill's analysis has been experimentally

verified by Elder [10]. In the present application, the solution corresponds

to a forced stratified channel flow. In the limit of zero stratification:

Returning to dimensional variables, equation (7) can be rewritten in the

classical form:

v « - (h/ dPR) 1/2 w

2 (l-C2

) (7)

v [ l-(z/d)

2

] ( 8 )

- 17 -
Here, y Is the viscosLty of the air, and the solution for u has been added.

For large strat Lf Lcat Lon, w is not small. As an example, for a stable

stratification - T

q

of 3°C, with the reference temperature T

q = 300°K and a

slot half width of 10 cm and wLdth to heLght ratio d/h = .01, w = 5.3. For

values of a) > 3, equation (6) simplifies to the form:

v = - (h/dPr)^^ {e ^ sin[w(l-<;)] + e ^ sin[to( 1+C) ] } (9)

3n

EquatLon (9) represents a vertical flow that has effectively ceased except for

a boundary layer of thickness (w) * near each wall of the slot. Thus, for a

given pressure gradient, there is much less vertical flow in the presence of

stratification than in Lts absence. ThLs can be seen more dramatically by

calculating the vertical volume flux of air per unit of length. The vertical

flux in dimensional variables is given by:

d

/ vdz = -d

f (to ) = —

8w

2d

3y

3

3

sinh(2to) - sin(2u>)

2 2 -sin (to) + cosh (to)

( 10 )

This should be compared with the horizontal volume flow per unit of height,

which is:

f

d

A ~2 d3 3p

j ud2 = _ _ _

-d (ID

Clearly, the function f(w) is a measure of the effectiveness of the pressure

gradient In producing a vertical flow. The function is presented in table

1. The decrease in effectiveness with increasing stratification

(increasing to) is quite obvious.

- 18 -
The final stage of thLs part of the calculation Is the determination of

the pressure in the slot. The pressure distribution is governed by the

requirement that mass be conserved in the slot. Let the quantity Q"(x,y) dA

be the rate at which fluid is Introduced by some external agent into the slot,

where dA is the element of surface across which the fluid crosses. This may

be an inlet or exit from a ventilation system or vent, or a cutout in an end

wall. Using equations (10) and (11), the conservation of mass yields the

following equation for the pressure:

There are three situations covered by equation (12) which are of

interest. First, if there is no opening into or out of the slot, then

Q" = 0. Second, if the flux through the opening is specified, then Q" is a

prescribed function. One such case of practical interest is a small opening

at x = x

Q

, y = yQ

which a total flow rate Qq

is specified, the dimensions

being small compared with the length or height of the slot. Then, Q" is given

by:

Here, 6 denotes the Dirac delta function. Finally, if the opening is large

and the pressure is specified at the opening, then it is more convenient to

consider the boundary of the opening as a boundary of the slot along which the

pressure is specified. Then Q" = 0 as before over the interior of the regLon

of interest. However, the solution now must be obtained over the rectangular

slot, with the correct pressure being specified at the open edge. The second

( 12 ) dX dy d

Q"(x,y) = Q

q

6(x-X

o

) 6(y-y

Q

) (13)

- 19 -
and thLrd cases are complementary In that the flow Ls specified and the

pressure is calculated In the second instance; whLle the pressure Is specified

and the flow is calculated In the third.

4. THE SLOT PRESSURE DISTRIBUTION

The starting point for the analysis Ls equation (12), with Q" = 0. The

boundary condLtLon at the closed end of each slot is (see equation (11) and

f Lgure 4) :

(A/2, y) - 0 (14)

At the open end, the pressure must be compatible with the end void pressure at

t~ K that height. If the end void pressure at the n

c slot is denoted by pv (n,y);

then the boundary condition at the open end is:

p(-JL/2, y) = P v

(n, y) (15)

At the bottom, since there is no flow through the floor of the hold, the

boundary condition is (see equation (10)):

(x, 0) = 0 ( 16)

Rather than consider the geometry of the air gap at the top of the hold

and its interaction with the upper boundary of the slot, it is more convenient

to note that most cases of practical interest correspond to values of w (see

equation (6)) such that f(w) << 1. If i/h is of order unity, then equations

-20-
(12) and (14) imply that, away from the top or bottom of the slot, p depends

only on y. Let Qn be the total flow of air drawn through the n

c, ‘ slot. Then

equation (10) and (12) imply that the pressure distribution over most of each

slot is given by:

P = I

M0

ny/dXf<“n ) (l7>

Now the same relation must hold in most of the end void, away from the

top or bottom. This means that the total flow, Q, drawn through the hold by

the ventilation system can be related to p by the formulae:

Q = N

E n=o

N

3

Z_ d

J l f(w ) n=0 n n n

(18)

The sum in equation (18) is assumed to extend over all slots and the end

void. Eliminating the pressure from this expression yields the result:

3

N

3

Q„ = Qi <Pf(u )/ Z <r<t f(u ) n n n n n-o n n n

(19)

Equation (19) is extremely important in what follows. It permits the

flow in each slot to be related to the total flow drawn through the hold, Q.

Thus, since Q is a prescribed system parameter, Qn can be determined in

advance as a function of the hold geometry and stratification. Physically,

equations ( 1 7)— ( 19) mean that the stratification completely suppresses

horizontal motion everywhere except near the top and bottom of each slot and

the end void. Equation (12) then implies that the horizontal motion is only

important in layers of order l^/ fw

n

) in height near the top and bottom. The

details of the motion near the top are of no interest. The only thing that

-21 -
matters is that the ventilation air enters there. The bottom horizontal

motion must be calculated because it determines the pickup of evaporated spill

material. However, it can now be calculated as if the slot were semi-infinite

in height with the boundary condition as y

00 given by equation (17).

To carry out the calculation, it is appropriate to proceed more

formally. Let the pressure in the n

1"^ slot be made non-dimensional as

follows:

3p yn ~N P = T T p(x,y)

d /f(w ) n n

x = x/i ; y = y/i /f (w ) n n n

( 20 )

Then the boundary value problem can be stated in the form:

2~ 2~ A±P

0

3x

2 8y

2

i£

8x

(1/2, y) = 0

^ (x,o) = 0

3y

( 21 )

Lim p(x,y) = y

y + 00

P(“l/2, y) = P v

(n, y)

-22-
Since the dimensionless end void pressure p^(n,y) Is unknown at this point; ii

is desirable to seek the solution In a form which displays the dependence

on p^(n,y) expliclty. This can be done with the aid of a Greens function

G(x, x

q

, y, y ) , defined as the solution of the problem:

2 2

3 G 3 G p \ p t ~ \ + = «(*-*,) «(y-y

0

) 3x 3y — (1/2, y) = ~ (x, o) = 0 ( 22 ) 3x 3y

G(-l/2 , y) = Lim G(x,y) « 0

-* 00

The quantity 6 in equat Lon (22) denotes the Dirac delta function. Introduce p

and G as defined by equations (21) and (22) into the divergence theorem in the

form:

/(p AG - G Ap) dx dy = 4

G O J (5 |£ . G |£_) ds

3n 3n o

o o (23)

Here n

Q denotes the outward pointing normal to the closed contour composed of

the slot boundaries and a fixed large value of yQ. Now letting yQ

•+ 00 and

using equations (21) and (22), a formal solution is obtained for the slot

pressure p(x,y) as:

oo P(x,y) = - / Pv

(n,y) (-1/2, y

Q

; x,y) dy

Q (24)

o o

In order to make equat Lon (24) useful, it Ls necessary to determine G

and p^. The solution for G is independent of p^, and only involves the slot

geometry. The solution for p^ will be obtained in the next section. The

-23 -
Greens functLon C(x, y, , y ) can now be obtaLned with the aid of a sequence

of conformal mappLngs. The steps in the sequence are (see figure 5 for

sketches of the mappings):

i) t = sin ( ttq )

C = x + iy

This transforms the slot into a half space with the open end of the bottom

at t = -1 (figure 5b).

it) = x + 1

This moves the open end of the bottom to the origin (figure 5c).

ill) w = e + In = (t

x

) 1/2

This converts the slot into a quarter plane with the open end on the positive

imaginary axis (figure 5d) . Thus:

1/2 W = £+iq=l + sin (tt£)

« ((a+ JT77 )/2)I/2

1 - U/a

2

+ b

2 - a)/2} 1/2

(25)

a = 1 + sin (irx) cosh (iry)

b = cos (ttx) sinh (n y)

-24-
I

The Greens functLon can be written down immediately in the W plane. A

solution is required with a logarithmic singularity at a point x = x^, y = y^

which vanishes for £ = 0 and whose normal derivative vanishes for q = 0. The

solution must be odd in £ and even in q. The result is readily obtained in

complex form as:

Equations (24) and (26) constitute the solution for the pressure in the

slot once the end void pressure is known. For later use, it is necessary to

compute the pressure gradient along the bottom of the slot. This calculation

requires considerable care, due to the nearly singular nature of the integral. The result, after considerable algebra, is:

G + iJ = ~ {log(W-W

Q

) + log(W-W

Q

) - log(W+W

Q

) - log(W+W

Q

)}

(26)

~3p , x cos (ttx) Tr (x,o) = K (x)

3x

(27)

{cosh (iry ) - l}

1 ^ 2 dy

o o

o sin (ttx) + cosh ( iryo

)

- 25 -
5. THE END VOID PRESSURE

The final stage Ln the determination of the large scale motion Is the

calculation of the end void pressure distribution. In order to proceed, It is

necessary to assume that the end void can be treated in the same manner as the

slots between container stacks, even though in many applications the relevant

void width d„ and length l are such that the ratios d /l and d /h are not V

u v V v v small. If these parameters are assumed to be small, then equation (12)

applies, with x now measuring horizontal distance along the end void. The

major difference between the slots and the end void lies in the appearance of

non-trivial sources and sinks Q"(x,y) in the end void.

n The slots between container stacks are narrow compared with the length or

height of the end void. Hence, the fluid issuing from them can be represented

as line sources of fluid in the form:

<Wn) = %

6 (x-xn

)q n

(y) (28)

Here Qn is the total flux issuing from the n

1

" slot, as given by equation

(19), 6 Is again the Dirac delta function operating at the horizontal location

of the n

L slot, and qn

(y) determines the distribution of flow with respect to

height. The distribution function q n (y) is normalized so that:

00

/ q n (y)dy = 1 o

-26 -
The air extraction system is assumed to have physical dimensions which are

small compared with the dimensions of the end void. Hence, Lt can be

represented as a delta function sink of strength Q, since It exhausts all the

air drawn into the hold. The geometry is sketched in figure (6).

The pressure in the end void is then determined by the solutLon to the

following system of equations:

+ Hu ) A = 4 \ Q"(x,y)

3x 3y dv

N

Q"(x,y) = E S(x-x

n ) ^(y) “ Q<5(x-xv

)6(y-y

v

) n=l

(29)

If ( x »°) “ 0

Lim p (x,y) =

-| hQ

oy/d^v

f (u>v

) y

+ oo

Here (xy , yv

) is the location of the air extractor, and Qq

is the flow which

originates in the end void, as determined by equation (19) with the void

geometric parameters.

The solution procedure is similar to that employed

section. The equations are made non-dimensional in the

in the previous

form:

-27-
3 u x

o

p = j

~2 p(x,y)

V

X = x/A ; y V

d_p + i

rf «

3x

2 3y

v v

N Q

" Z 7T

n=l Qo 6(x-x

n

) q n (y)

N Q + 1 + z \ n=l

6(x-x

v

)5(y-y

v

)

(± 4, y) = (x,o) = 0

3x 3y

Lim p(x,y) = y

y + oo

(30)

A Greens function is again introduced, this time solving the following system

of equations:

= 6(x-x

q

) 6(y-y

Q

)

3G , 1 ~ ~ — (x = ± y, x , 3x

^ o — (x, 0, x , y ) = 0

3y

°

(31)

Lim G (x, y, x

q

, y

Q

) = y

y

oo

Substitution of equations (30) and (31) into equation (23) then yields the

result:

- 28-
( 32 )

/ n

N Q

'P(x,y) = 1 + £

V n=l vo G (x, y, x

v

, yv

)

N Q - E 77^- / q (y ) G(x, y, x , y ) dy

_ 0

J ^n J o n* J o o n=l o o

The solution is completed by specifying G and qn

. The Green's function is

determined by noting that the first of the transformations employed In the

previous section maps the end void into a half plane. The solution for G is

then readily obtained as:

G + iJ = — {log(x-T

o

) + log(t-T

o

)}

/v x = sin(irs) = sin(nx) cosh(iry) + i cos(irx) sinh(iry)

(33)

/V <\> «v rv t = sin(irx ) cosh(iry ) + i cos (irx ) sinh(iry )

/V /V /V x = sin(xx ) cosh(iry ) - i cos(ttx ) sinh(iry ) o o o o o

The flow distribution functions, q n , are in reality not arbitrary, but

must be determined by the condition that the pressures as computed from

equations (32) and (33) lead to the same flows when the solutions given by

these equations are substituted into equation (24) and equation (24) is

differentiated to obtain the flow out of each slot. In general, this leads to

a system of N integral equations for the void pressure at each slot. The

solution can be approximated with reasonable accuracy, (i.e enough accuracy to

evaluate equation (27)) by noting several points. First, the fact that the

total flow issuing from each slot is known implies the constraint given by

equation (30) on qn

(y)» Second, the flow should be greatest at height

-29-
y = y^ , since that Is the level at which the aLr Ls drawn out. Third, the

flow should ultimately decay exponentially with distance away from its

maximum, since the Integral equations have the Greens functions for kernels,

and the Greens functions all exhibit this type of decay. Finally, examination

of equations (27) and (32) shows that the pressure cannot be sensitive to

details of the shape of q

n (y). Hence, in the spirit of Carrier [11], the

following form for q^(y) * s postulated:

V50 = exp {-it y-y

[2-e-^vJ

(34)

Equation (34) is consistent with all the points mentioned above. It also

permits the integral in equation (32) to be evaluated in closed form. The

result, after some extremely tedious algebra ls:

/ N Q \

p(x,y) = 1 + Z -pp

) G(x,y

; \yYy

) \ n=l v

o /

N + E n=l

Qq 1 Q

° 2n(2-e

-1T^v)

4

{ 2 L(t ) 3=1 J

log [ ( 1-a j)

2 + b2.]}

- 30-
+ log

2 2

(1-a r + bj

, -iry \ 2 ,2

(e v - a ,) + b .

1 e

_7ryv , 2 a

j

. 2 2.

108

(a

j

+ V

» - V

2 +

(1 - a^ e^v)^ + (bj e^v)^

( 35 )

-Try ,

e v b

(a

Til f + b?) (

arctan

-Try a

.i ~ 6

j J

j

- arctan

a

j - 1

i. = -e

1^ cos [tt(x + x )]

1 n

b. = sin [tt(x + x )]

1 n

a, = e ^ cos [ti(x - x )] 2 n

b 0

«* e

*y sin [tt(x - x )] 2 n

—iry a 0 = -e J cos [tt(x + x )]

J n

rv

b 0 = e sin [tt(x + x )] j n

a^ = cos [ir(x - x^)]

b^ = e Uy

sin [tt(x - x

r

) ]

-31 -
6. THE BOTTOM MOTION

When the vertical distance above the hold bottom becomes comparable to

the slot width, the flow pattern departs from that calculated in previous

sections. While the length l of the slot is still long compared with the

half-width d, the vertical scale is now of order d since the downward flow

must terminate at the bottom. The boundary layers at the sides of each slot,

which carry the ventilation air downward, must spill out into the bottom

across the full width of the slot. The horizontal motion must also adjust so

that it can come to rest at the bottom.

To proceed, we consLder the dLmensional dependent variables introduced In

equation (2). Substituting these into the linearized form of equation (1)

(recall that Re ~ 0 (1)), the equations of motion become:

3 u _3v 3w

3x 3y 3z

0

i£ = 3x yAu

3p Pg

3y T

o 0 yAv (36)

i£ =

3z yAw

T

1 h

To

V

y pPr

AG

-32-
The geometry is shown schematically in figure (7). Equation (36) is to be

solved subject to the following boundary conditions:

u(x, y, ± d) = u(x, 0, z) = 0

v( X, y, ± d) = v(x, 0, z) = 0

w(x, y, ± d) w(x, 0, z) = 0

(37)

0(x, y, ± d) = 0(x, 0, z) = 0

Finally, for y >> d, the solutions for u, v, w, and 0 must merge smoothly with

those obtained in sections 2-5. This statement will be made in a more

quantitative fashion below.

The solution procedure is based on explicitly recognizing the differences

between the four relevant length scales in the problem. These scales are, in

decreasing order of magnitude:

1.

) The slot length %

2.

) The scale height for large scale motion (A /f (to)

3.

) The slot half width d

4.

) The slot wall boundary layer thickness (d/w)

The slot bottom region is now divided into two wall boundary layers and an

interior region. In the interior region the dependent variables are expanded

in an ascending series in the parameter (d/Jt) of the form:

-33-
MQ,

p

=

2 ,3 d /f ( a)

( P (x ,0) + A2 p*(x,Y,Z) + ....}

u =

3 Qo Ad/f(w)

(u*(x,Y,Z) +

v =

-| (j) (v*(x,Y,Z) + . . . .}

Jld/f (to)

W = -r- 3 y

o Wf (w)

( 4) (w*(x,Y,Z) + }

( 38 )

0 = T . PQ , , (4) {e*(x,Y,z) +

cgd 2 dvfcsr r

x = x/i; Y = y/d; Z = z/d

Note that in equation (38); p(x,0) is the pressure obtained from the calculation of the large scale motion in section 4. The velocity components and

temperature are scaled to ensure consistency with the large scale motion and

with each other. Substitution of equation (38) into equation (36) and

2 ignoring terms of order (d/i) yields:

3u*

3x

+

3v*

3Y

0*

2 2

3 u* 3 u*

2 +2 3Y 3Z

2 2

3 u* 3 w*

2 +2 3Y 3Z

(39)

-34-
Since the driving force in equatLon (39), —^

(x,0) is known, the velocity

3x

component in the direction of the slot, u*, can be obtained separately from

the other variables, as the solution of (recall the second of equation (8)):

3Y

£ +

^ (5,o)

3Z 3x

u* (x, 0, Z) = u* (x, Y, -1) = u* (x, Y, +1) = 0 (40)

Lim u* (x, Y, Z) - (x, 0) (1-Z

2

) y > oo 3 X

To proceed, u* is written as the sum of the large scale motion near the bottom

plus a correction.

rs* u* = “ 7

(x, 0) (1-Z

2

) + u 3x

a 2- a 2-

2 2

3Y 3Z

u (x, Y, -1) - u (x, Y, +1) = 0 (41)

u (x, 0, Z) = 4^ <*. °> a-z

2

) 3x

Lim u = 0

Y -* oo

The correction u can be expressed in terms of a Green's function

G(Y, Z; Y , Z

Q

) in a manner analogous to that described in previous

sections. The result of the calculation is:

-35-
u(x, Y, Z) = (x,0) / dZ

Q

(l-Z

2

) ||- (Y, Z; 0, Zq

) 3x -1

G + iJ = — {log (<M>0

) - log (4>-4> o

)

}

<\> = sin (ttX) ; X = Z + iY ( 42 )

<j> = sin (ttX ); X = Z + 1Y

o o o o o

) = sin (ttX ); X = Z - 1Y

o o o o o

The final expression for u* can be rewritten in a more convenient form for

3G computation by employing the Cauchy-Riemann equations to eliminate -r~— • The

<3Y o result is:

u* (x, Y, Z) = - —^ (x, 0) {- (1-Z

2

) - / dZ Z J(Y, Z; 0, Z )}

3 x -1 o o

T 1 „ ( I(Z,Y)

J = - arctan

{ R

-(Y , 0) }

if R(Z,Y) > R(Z ,0); (43)

J = — { tt - arctan

I(Z t Y)

R(

Q

’0 ) - R(Z,Y)

if R(Z ,0) > R(Z,Y).

The quantities I and R are given by:

-36 -
( 44 )

R(Z,Y) = sin (-{- Z) cosh (-| Y)

I(Z,Y) = cos (-| Z) sinh (j- Y)

Note that u* as given by equations (43) and (44) has the form:

u* = - (x, 0) U(Y, Z) (45)

3x

Thus, the velocity profile at each axial station x = constant has the same

"universal profile" U(Y,Z). This profile is displayed in figure (8).

The x dependence can be factored out of all the variables in equation (39);

the resulting decomposition being given by:

w*

e* = (x,

3x

p* - ^

(X,

3x

0) V(Y, Z)

0) W(Y, Z)

0) 0(Y, Z)

0) P(Y, Z)

(46)

The prof Lie functions V, W, 0, and P satisfy:

-37 -
av _aw 3Y +

az

u

ap

az

2 2 aw aw

2 +

2 3Y az

ap

aY

0

a 2 v a^v

3Y

2 az

2

2 2

, 4, „ a 0 , 30

(w ) V = r- + T- 3Y az

( 47 )

In order to proceed further, Lt Ls necessary to consider the dependence

of the solutions on uj. In particular, the structure of the wall boundary

layers (of thickness (w) * on the slot half-width scale) must be determined

(see figure 7). The boundary layer structure can be found from equation (47)

without loss of generality; since the x dependence factors out in the form

given by equation (47) everywhere in the slot bottom. Symmetry considerations

then permit attention to be confined to the wall layer near Z = -1, the layer

near Z = +1 being identical. In this region:

V = a) V (C, Y)

W = W (C, Y)

0 = a) 3

0 ( 5 , Y) (48)

P = u) 3P (e, Y)

C = a) (Z + 1)

-38-
Substitution of expressions (48) into equation (47) and keeping the leadLng

order terms in u) leads to the wall boundary layer equations Ln the form:

(49)

3P

3Y

At the wall, C - 0, the velocity components and the temperature perturbation

must vanish (the latter due to the assumed equilibrium between container

stacks and hold stratification). As £

00 these solutions must match the

expressions for V, W, 0, and P (which have not yet been found) in the interior

of the slot bottom region. For the present, we assume only that all quantities are bounded in the interior, as C + ®.

Equation (49) may be readily solved by noting that, from the second of

these equations:

P = P(Y) (50)

Although P(Y) is as yet unknown, V and 0 may then be found in terms

of P(Y) as:

-39-
^ ^ p 0 =

"3Y

^ 1_exP C -^/^ 2 ) cos (C/*^2)}

V = - — exp (-;//2) cos U//2)

( 51 )

Note that as C + 00 (l.e. as the interior of the bottom region 0 Ls approached)

0 -*• dP

dY

W > (2)

-1/2 d P

2Y

(52)

V * 0

In order for the interior functions (the solutions to equation (47)) to

have proper scaling wLth respect to w, they must be consistent with equation

(52) as Z - ± 1. This can be achieved by rescaling as follows:

V(Y,Z) = (o))"

1 V

I (Y,Z)

0(Y,Z) = (w)

3

0

x(Y,Z) (53)

P(Y,Z) = (w)

2 Pj( Y,Z)

The leadLng terms Ln the interior equations then become:

-40-
aw

az

ap.

'az' - o (54)

3P

I TT " G

i " 0

From the second of equation (54) and equation (50)

P

x = P(Y) (55)

The last of equation (54) is now consistent with the first limit in equation

(52), yielding the result:

0 = — (Y)

I dY K J (56)

Finally, the first of equation (54) yields:

W = / UdZ

o

where

1 O

* u = ± (1-Z^) - / dZ

Q

Z

Q

J( Y, Z; 0, Zj

-l

(57)

(see equation (43)).

The transverse profile W is displayed in figure 9. Note that U is

symmetric in Z, so W is anti-symmetric. The remaining unknown P(Y) Ls

-41 -
determined by requiring that W(± 1, Y) be cosistent wLth the matching

condition given the second of equation (52). Thus:

= - /2 / U(Z,Y)dZ

0

1

( 58 )

This equation can be integrated once with respect to Y, using the

dP value — (0) = 0 to ensure that the vertical velocity in the wall layer

dY

vanishes at Y = 0 (see equation (51)).

The most important results of this section are equations (43) and (57),

which yield the profiles for the two principal velocity components in the

bottom region. These profiles will now be used in the calculation of the

vapor pickup in this region.

The calculation of the vapor pickup requires a solution for the vapor

concentration gradient at the bottom of the hold. In order to proceed, It Is

necessary to recall the spill scenario postulated in section 2. It is now

further assumed that the bulk of the pickup takes place along the bottom, but

outside the wall boundary layers. The concentration C(x,y,z) then obeys the

equat Ion:

operator; and u and w are the velocity components determined in the previous

7. THE VAPOR PICKUP

(59)

Here D is the diffusivity of the spill vapor in air, A is again the Laplacian

-42-
section. At y=0, the concentration is assumed to be C

Q

, the equilibrium vapor

pressure at the temperature corresponding to the hold bottom. Outside the

layer, there is no vapor; C=0.

We now non-dimens ionalize the velocities and coordinates as in equat Lon

(38). The concentration equation (59) then takes the form:

u* (x,Y,Z) + w* — (x,Y,Z)

9x

_L_ A c R*S 2 e c

2 2

A c ^

2 ^ "

= 0

2

+ 0

2

3Y 3Z

(60)

3Q

R* = —

2Jtd/f(w)

° (<L) (A)

K vJ V JT

S - v/D

c

Equation (60) must be solved subject to the boundary conditions:

C(x,0,Z) = C

q

Lim C(x,Y,Z) = 0

Y + oo

(61)

The solution procedure employed Ls a generalization to three dimensions

of that used by Lighthill [12] in obtaining his heat transfer formula. Recall

that equations (45), (46) and (57) allow u* and w* to be expressed as:

-43-
( 62 )

w* = (x ,0) W(Y,Z)

3x

Following Lighthill, the vertical dependence of velocLty profiles Is approximated by a linear function.

This approximation may be justified in several ways. First, when the Schmidt

number S

c » 1, it is rigorously true that this simplification yields the

asymptotic solution for the concentration profile. Lighthill has shown that

In the case of heat transfer, the approximation works quite well for Prandtl

numbers of 0.7, corresponding to air. In the present application, S c is

usually in the range 1.5 < Sc < 2. For this range of Schmidt numbers, the

concentration field is largely controlled by the velocity profiles near the

bottom. Inspection of figures (8) and (9) shows that the velocity profiles

are fairly linear in this region. Finally, it should be noted that only the

wall concentration gradient is required, not the whole concentration

profile. Such information can be (and often is) obtained using much cruder

profile information than will emerge from this calculation.

It is convenient to express the velocity components in terms of a stream

function 4>(x,Z) defined as:

W(Y,Z) = W (Z)Y, W ’ o o

V 3Y' Y=0

(63)

Kx,Z) = - — (£,o) W (Z)

3x

( 6A)

-44-
The velocity components are given by:

u* y H ;

w* = - Y

9 ^9x

The vertical coordinate is now rescaled as follows:

X = (R*S ) 1/3Y

e c

( 65 )

( 66 )

The concentration equation now becomes:

•!

jhp 9_C

3Z

3x

dip dC

a;

3Z (67)

The solution of equation (67) depends crucially on the observation that

curves of constant i|; represent the trace of the streamlines calculated in

section 6 on the bottom. These streamlines originate in the wall layer at the

side of each container stack. Let s denote distance along each streamline

with the origin at the point where the streamline emerges from the wall (see

/v figure 10). Using s, i|> as independent variables in place of x, Z; equation

(67) becomes:

q (s ,\p)

( 68 )

-45-
At s=0; the ventLlation air has just entered the bottom regLon; hence

C=0. At A=0; C=C and C -* 0 and as A

00 from equatLon (61). This Ls a o relatively straightforward problem. To proceed, we introduce a modified

streamwise variable £ defined as:

s

5 = / ds/q(s,\J>) (69)

then

n 3C

(70)

Introducing laplace transforms with respect to equation becomes:

pAC

00 C = / e"P? ca,A)d£ (71)

0

C

ca.O) = — ; C 0 as n > «

P

The solution for Z satisfying equation (71) is readily found [13] to be:

C = -2- T (2/3) (3)

2/3

Ai ((p)

1/3

X) (72)

Here Ai is the Airy function, and T the Gamma function as defined in reference

[13].

-46-
Although Inversion of £ to obtain the concentration profile would be a

formidable undertaking, the problem becomes tractable if only the wall

concentration gradients are required. The mass flux picked up at each point

by the ventilation system, m, is given by:

= _ £ (

\ 1/3 3C

d (Re

S c

) 3X

(73)

From equation (72), the laplace transform ro is readily computed as:

. D . .1/3 . .1/3 T(2/3) , ,-2/3 m -

p

(R

e

s

c

) c

o

<3) mm (P) (74)

Inverting equation (74) and recalling the definition of C from equation (69),

the mass flux becomes:

m

3R S o r( 1/3)

s

{ / ds/q(s,\|0 )

1 0

f( 1/3) = 2.67894...

(75)

Equation (75) yields the pickup at each point in a given slot. The

quantity actually desired is the total mass pickup. The total mass pickup in

a slot M is given by:

d l

M = / dz / dx m

-d 0

1 1 = 2d l J dZ / dx m(x,Z)

0 0

(76)

-47-
Now let s and n be coordinates along and normal to a streamline ^

-

constant. Then, from equations (75) and (76):

M = dD£ C

(3R*S ) e c o r( 1 / 3 )

1/3

g // dsdn {/ ds/q(s,40>

0

( 77 )

Using the fact that dn = dij>/q, and d£ = ds/q, it Ls possible to carry out the

integral along streamlines to obtain:

M = 2M C (3R*S ) o e c 1/3

r( i/3)

pM .. 3 r c / . X i2/3

I dij> 2" L5(^)j

(78)

sfM

( ^ ) 5 M

(40 = / ds/q

0

Here s^(i|0 denotes integration over the entire distance along each streamline

from the point it enters the bottom until the time it exits into the end void

(see figure 10). Similarily, iJj denotes the maximum value of the stream

function, so that the integration covers all streamlines originating in the

wall layer.

At this point it is convenient to recapitulate the overall calculation

procedure. The first step is the determination of the flow assigned to each

slot and to the end void. This is given by equation (19), which yields the

total flow in each slot, Qn , as a function of the total flow drawn through the

hold, the hold geometry, and the degree of stratification. The next step is

the computation of the pressure gradient along the bottom of each slot

-48-
containing spill material. This pressure gradient controls the development of

the spill material boundary layer, and hence the rate at which spill material

is picked up by the ventilation system. The necessary result Ls given in

equation (27). Note that this formula, in turn, requires a knowledge of the

variation of the void pressure p^ with height at the open end of each slot In

question. The void pressure at any point is given by equation (35), which

requires only the information already obtained from equation (19). With the

pressure gradient along the slot bottom now determined, the velocity distribution near the slot bottom are given by equations (43-46) and (57). These

results are then used to get approximate simplified formulae, equations (63)

and (64), which are actually used in the calculation of the rate of pickup of

spill material. These latter formulae express the velocities near the bottom

of each slot in terms of a "bottom stream function" \J/. Given the

quantity i|>, the magnitude of the velocity gradient at each slot bottom, q, can

be determined from equation (68). Finally, given q and ip, equation (77)

• yields the total mass pickup in each slot M. These results, summed over all

the slots containing spill material, yield the total mass per unit time

extracted from the hold by the ventilation system. The computer program which

executes these calculations is of necessity quite elaborate. The following

sections describe the overall program layout, the principal subroutines,

sample results and a complete listing.

-49-
8. NUMERICAL PROGRAMS

Two main programs were written. The larger one carries out the numerical

calculations of the contalnership hold ventilation model; while the other

plots selected pressures and streamlines. Although the plot program uses

CALCOMP emulation subroutines, the program is sufficiently specialized to our

computer that it would have limited interest. It will not be described.

8.1 Main Numerical Program

The main numerical program controls the entire calculation, i.e., all

subroutines are called from the main program. There are eleven of these. In

addition, there are two external function routines. These are called from

subroutines. A simple block diagram of the program is shown in figure 11.

The first two subroutines, INTEG1 and INTEG2, carry out numerical

integrations related to the flow at the bottom of a slot. In the dimensionless variables used, these are geometry independent. PRINT then tabulates the

geometry independent dimensionless slot bottom quantities. Since they do not

depend on input data they are carried out once and the results saved for use

in all variable geometries to be calculated.

The program is arranged to calculate multiple geometric or thermal

cases. The number of these is read (NGEOMS) and a loop entered between CALL

INPUT and the end of the main program. INPUT reads, from the input file or

device, logical unit 5, the geometric and temperature values to be used. At

this point the main program performs two scaling operations, one for the main

- 50-
transverse space (which will be referred to as the "main voLd" or simply the

"void") and the second for the gaps extending fore and aft between stacks of

containers (referred to as "intercontainer slots" or simply "slots"). These

operations correspond to the quantities to and f(w), defined in equations (6)

and (10). The void and the intercontainer slot volumes are assumed to be the

only unoccupied space in the hold. For ships with wing tanks, the main void

is the volume on the framed side of the transverse bulkhead between the

bulkhead and the container stacks. Where there are no wing tanks, the ma Ln

void is increased in length to account for the volumes between the outermost

container stack and the side of the ship. The scaled lengths of the void and

slots are then compared to their respective widths to see if the assumptions

of the model are satisfied, namely that the scaled widths are less than theLr

lengths. Actually, the wLdths should be much less than the lengths. If

either of these scaled lengths is less than the respective widths, program

execution stops with an appropriate message.

The air issuing from the slots into the main void could be a problem to

the model if the flow were strong enough to have a jet-like character. In

addition, if the slots were spaced close enough together, after scaling,

compared to the width of the void they might interact violating another

assumption of the model—that the slots are widely spaced. For some actual

ship hold geometries this condition may, in fact, be violated. However, the

void is not a simple, smooth sided space as assumed by the model but, rather,

a complex of interconnected volumes formed by the framing of the bulkhead. It

is further broken up by supports for the containers. The structure provides

considerable surface for heat exchange and can break up and separate any

jets. For practical purposes there seems no way to treat the void correctly

- 51 -
in any detaLl. Accordingly, It was decided to treat It as a simple smooth

space as per this model, recognizing that this was questionable. Having made

this decision, the scaled slot spacing to void width ratio Is computed and

printed. If this ratio is not small a warning Is also printed.

If the transverse location of the suction aligns with one of the slots

there will be numerical problems. Should this occur, the suction Is arbitrarily moved away from the slot 1/2% of the void length. This is a distance

less than the resolution scale of the overall calculation. Another check

looks at the location of the (single) suction to see if it Is very near the

tank top, Y=0. The calculation assumed it is not. If it is, it is arbitrarily moved up 5 cm. Next, the slots numbers, for the slots furthest and

closest to the suction, are determined for later use in plotting pressures and

streamlines. It turns out that, even for very small stable strat Lf icat Lon

temperature differences, the differences in the flow in the several slots Ls

very small, hence only selected results need be displayed to adequately show

the flows.

Having completed these housekeeping details, the calculation proper

begins. The main calculation is a large loop, passing once around for each

slot. Within the loop the slot bottom flow is calculated, SL0T1 , and

streamlines traced, STRM1. The mass pLck-up is then obtained for the separate

stream tubes by integrating along the streamlines, PICUP1. Finally, the total

mass evaporated from the slot is obtained by integration over all the stream

tubes, also PICUP1. This is repeated for each slot and the total mass

evaporated obtained by summing the contributions from each slot. In tracing

the streamlines, a matrix, ZPSI, giving the transverse location, z(x), for

-52-
various constant values of slot bottom stream function is formed. This is

used in calculating the stream tube integrals and is unique to any given

slot. Since the matrix is over-written with each pass through the loop, the

ZPSI matrix for two slots—the one closest to the suction and the one furthest

from it —are saved as a convenience for later plotting. Upon completion of

this calculation for all the slots, results are printed. PRNT1 tabulates the

dimensionless slot velocity and stream functions. The axial velocity near the

bottom of the slot, USLOT (u of equation (43)), is a function of axial

distance along the slot and differs for each slot. This is the axial velocity

in the slot just above the boundary layer at the slot bottom. The slot bottom

boundary layer velocity approaches USLOT asymptotically with height. The slot

bottom stream function defines the track of the air as it passes over and

evaporates the spilled liquid assumed to be spread over the hold bottom.

Stream function values are tabulated for each slot—the value of the stream

function, PSI, for fixed locations along the length of the slot (rows) and

fixed locations across the width of the slot (columns). Next the dimensionless mass pick-up for each slot is printed together with the corresponding

dimensionless slot location.

The intent in developing this program was to be able to apply the results

of the calculation to specific chemicals. Calls are made to INPUT1 where

vapor pressure, temperature and diffusivity data are read for the number of

chemicals specified in INPUT and NCHEMS. The total mass pick-up, in dimensional form, is computed for the hold ventilation rates read by INPUT2. Note

that the mass pick-up varies in an almost trivial way with ventilation rate,

the l/3rd power, so that an eight fold increase in ventilation doubles the

mass pick-up. The vapor concentration in the exhaust air is also computed for

- 53-
SL0T1

SLOT1 is a very short, simple subroutine but, through it use of the

external function GRAD, which in turn employs the external functLon PRESS,

Lnvolves a faLrly complex program structure. The axial flow in a slot depends

on the axLal pressure gradLent. SL0T1 determines the characteristic magnitude

of the velocity USLOT, which is the same as —

3, near, but not quite at the

9x

slot bottom, the first term on the right of equation (45). This term Is

determined by equation (27). The boundary layer-like analysis of the local

flow at the very bottom of the slot detailed in INTEG1 and INTEG2 is scaled to

approach asymptotically the velocity USLOT as height increases away from the

slot bottom.

GRAD

Subroutine GRAD carrLes out the integral, equation (27), which gives the

pressure gradient just above the hold bottom, 3p/3x(x,0), for each slot

(needed by subroutine SL0T1 as discussed above). This integral requires the

vertical pressure gradient In the end void at the location of the slot, for

each slot. This is found by a call to subroutine PRESS. There are two

noteworthy aspects of this calculation:

(1) The integrand of the integral carried out in GRAD has an

integrable singularity for x = -1/2 (the slot void intersection) and y=0. Rather than carry out the rather messy

algebra to Integrate this formally and provide a program option

- 56 -
subtracting the value of UB at Y=0, which is 0, and multiplying the result by

20. Figure 12 shows UB for Z=0 as a function of Y. It is seen to be quite

smooth Indicating that this approximation should be quite good. Examination

of figure 8 shows that this is also true for other values of Z.

INTEG2

INTEG2 forms the indefinite Integrals with respect to Z of the quantity

UB which yields W, equation (57) and of UBO which yields W

Q again using the

trapezodial method. W is shown in figure 9.

PRINT

This is a fairly straightforward tabulation.

INPUT

For each geometry, temperature difference (stratification) data is

read. Then hold geometry, as reflected by the main void and slot geometries,

is read. Several consistency checks are made: the number of slots must be no

more than 11 (dimension statement and format limit) and the number of x values

along the slot, NXSLOT, for which values will be computed must not exceed 30

(dimension statement limit). Also the spacing between slots times the number

of slots must not exceed the length given for the main void. Failure of any

of these tests results in termination of execution with an appropriate message

printed. The slot-void intersection locations are calculated placing the

slots symmetrically about the centerline of the void. The suction locatLon is

then read.

- 55-
SL0T1

SLOT1 is a very short, simple subroutine but, through it use of the

external function GRAD, which in turn employs the external function PRESS,

Involves a faLrly complex program structure. The axial flow in a slot depends

on the axial pressure gradient. SL0T1 determines the characteristic magnitude

of the velocity USLOT, which is the same as -^3, near, but not quite at the

3x

slot bottom, the first term on the right of equation (45). This term Is

determined by equation (27). The boundary layer-like analysis of the local

flow at the very bottom of the slot detailed in INTEG1 and INTEG2 Is scaled to

approach asymptotically the velocity USLOT as height Increases away from the

slot bottom.

GRAD

Subroutine GRAD carries out the integral, equation (27), which gives the

pressure gradient just above the hold bottom, 3p/3x(x,0), for each slot

(needed by subroutine SL0T1 as discussed above). This integral requires the

vertical pressure gradient In the end void at the location of the slot, for

each slot. This is found by a call to subroutine PRESS. There are two

noteworthy aspects of this calculation:

(1) The integrand of the integral carried out in GRAD has an

integrable singularity for x = - 1/2 (the slot void intersection) and y=0. Rather than carry out the rather messy

algebra to Integrate this formally and provide a program option

- 56 -
to use the alternative algebra, use was made of the fact that

the result is smooth as x = -1/2 Ls approached. Numerical

experiments for a series of values of x approaching -1/2 showed

that using x = -0.498 gave a result very close to the extrapolated value without encountering numerical problems, x =

-0.499 gave a value fairly close to the extrapolated one but was

suggestive of an approaching numerical problem, see figure 13.

(2) The integral is over an infinite range. The infinite integral

is completed by adding an analytically determined piece for

argument three to infinity to the trapezodial rule computation

from zero to three.

PRESS

The function PRESS is called from GRAD (and also from PPLOT) . It

computes the pressure at the intersection of the slot at transverse location

XS(N) with the main void using equation (35). This pressure depends on the

flows from all the other slots.

The calculation involves two pairs of two terms each and, as will be

discussed later, has some numerical problems since these terms involve small

differences of not very small terms. These problems appear to be large enough

to require the use of double precision on 32 bit computers. Figure 14 shows

the result of calculating pressure in single precision. A double precision

version of PRESS produced smooth plots, compare figure 14 with 20a. Numerical

values of the mass pick-up were about 4% lower when using the double precision

-57 -
pressure routLne as compared to s Lngle precision. Subroutines GRAD and PRESS

were originally written by Dr. D. Corley of Montgomery College and NBS.

STRM1

The value of the stream function in the Nth slot at a point X(I) axially

along the slot length and z(k) transverse to the slot center line is given by

(see equation (64)):

PSI (I,K,N) = USLOT (I,N) *WO(K)

where, as already discussed, USLOT is the characteristic velocity magnitude at

I this axial position in this slot as tabulated in SL0T1 and WO is the indefinite z integral determined in INTEG2, of UBO, the gradient of the slot bottom

velocity determined in INTEG1. The task, of STRM1 is to produce, from values

of the stream function at fixed geometric points, a table of coordinate pairs

(x,z) which correspond to fixed values of stream function. Thus, it is rather

analogous to a contour plot routine. The routine chooses an x from the

equally spaced ones for which USLOT and WO are known. It determines the PSI

value at this x and at the slot edge, z=l. Since flow descends along the slot

walls and flows out onto the slot bottom, . streamlines originate at the slot

edge. The streamline originating at this x is then followed toward x = -1/2.

At each fixed x between the initial one and the far end of the streamline a z

is determined, by linear interpolation between the fixed z's for which WO is

available, where the required stream function value is found. The result Ls a

table of z(x) for fixed PSI, ZPSI. Most streamlines terminate at z = -1/2

with a z between 0 and 1. However, especially for slots near the suction and

- 58-
suctions raised above the slot bottom, not all the flow goes dLrectly out the

end of the slot. Some of it may return to the slot s Lde wall and flow up the

wall to near the height of the suction where it finally leaves the slot.

Streamlines for this flow are shown schematically in figure 10. This flow

behavior is associated with a maximum in PSI for an x greater than -1/2. To

Identify the location of the end of a streamline it is useful to record the x

location of its end, either -1/2 or the x along the slot sidewall to the left

of the maximum PSI where the required stream function value occurs. This Ls

obtained by linear interpolation and recorded in XPSI(N). The matrLx of

values of ZPSI is returned to the main program and passed forward to PICUPI

but is not retained as successive slots are treated.

PICUPI

The mass evaporated from the spill pool at the bottom depends on the

velocity and path over the pool. The pick-up integral is given by equation

(77). The limits of the integral are set by the extent of the spill. It has

been assumed that the spill covers the entire hold bottom so the limits have

been made on the wall where the flow enters the slot bottom and either the

slot-void junction or the wall, if the flow returns there as described

above. It would have been a simple extension of the program to input the

extent of the spill pool, presumably less than the full length of the slots

and different for each slot, but this seemed to offer little advantage since

there is no information on the expected size or extent of a spill. As is done

throughout the program, the trapezodial algorithm is used for the integration. Once the separate stream tube integrals have been formed the total mass

pick-up for the slot is formed by integrating over the stream tubes.

-59 -
PRINT1

After all the slot quantities have been calculated the results are

tabulated. The values of USLOT are printed in columns for each slot with rows

corresponding to axial position X, along the slots. The columns are headed by

the dimensionless slot location. The stream function is then tabulated for

each slot with columns giving the transverse, Z, location and rows the axial

position, X. In this case, slot location is given in dimensional units. This

is followed by a short table pairing the dimensionless slot location with the

total, dimensionless pick-up for that slot.

INPUT1

In INPUT the counter NCHEMS was set. The main program now loops through

INPUT1 reading chemical data for as many materials as requested by NCHEMS.

The chemical data entered is the mass diffusivity and partial pressure data

using the "Antoine Formula" [14]. The data is then corrected for the case

temperature to give the saturation concentration and corrected diffusivity.

For each chemical, several flows can be used. The effect of flow is very

simple but this feature was included as a convenience. The number of flows to

be considered for each chemical is read, NFLOWS.

INPUT2

This simply reads the flow and returns. It is called as many times as

requested by NFLOWS.

-60-
PPLOT

The pressure at the slot-void intersection for the same pair of slots for

which streamline data was saved in STRMl is now calculated using the external

function PRESS. This is done separately rather than saving values computed

during the slot calculation to provide some additional flexibility Ln the

range of values computed. The cost in computing time is not great.

8.3 Input Data

A sample input data file is given in appendix 1. The example is for a

single case with a AT of 0.1°C. The length of the main void (in this case the

width of the hold plus twice the container length to simulate a ship without

wing tanks) is 51 m, its width (fore and aft dimension is 0.8 m and its height

20 m. The intercontainer slots are 12 m long (fore and aft), 0.1 m wide

spaced 2.46 m along the length of the main void. Since the slots are the same

height as the main void their height need not be input. Twenty points are

calculated along the length of the slots and there are 11 slots. The suet Lon

is placed at and a nominal height = 0.6 m and transverse location relative to

3 the center line of the main void of 0.0. The hold empty volume is 2407 m and

3 the void volume is 1920 m . One chemical is considered. The caption is for

this chemical; the caption is limited to one line of 80 characters, maximum.

Chemical data entered is:

- 61 -
Mass diffusLvity of spill vapor at 0°C cm^/sec

Molecular weight of spill chemical

MaterLal constant, B of Antoine's formula, °K

Temperature at which spill chemical has a vapor pressure of 1 atm J C

Reference temperature, C of Antoine's formula, °K

Ambient temperature at spill liquid, °C

O One flow is considered and the flow is 1.0 m /sec. The result of executing

the program (in double precision) wLth this input is reproduced in appendix 2.

9. NUMERICAL RESULTS

The purpose of this study was to predict the mass evaporation rate for a

hazardous liquid spill in a containership hold. Since no information was

available about the extent of such a spill, it was assumed that the liquid

would spread uniformly over the bottom of the hold (tank top). In practice,

there is almost always some trim to the ship so that liquid would tend to

drain to the aft end of the hold. A low point sump could collect some of the

liquid which might be advantageously pumped to a safe place thus reducing the

amount of liquid that would have to be removed by the rather slow evaporation

process. Rolling of the ship would tend to spread the spill across the entire

width of the hold. As discussed in the section of this report describing the

numerical programs, the spill was assumed to extend fore and aft along the

entire length of the intercontainer slots and athwart ships across all the

slots. However, no evaporation was computed for the main void comprising the

transverse space between the bulkhead and the end of the container stacks.

Unless otherwise noted the calculations were done for a hold containing twelve

- 62-
stacks of 12 m (40 foot) long containers and with eleven lntercontalner slots

0.1 m wide. The container wLdth was 2.36 m, giving slot spacing of 2.46 m.

The air temperature near the bottom of the hold was taken as 10°C. Dimensional mass evaporation rates are based on vapor pressure and diffuslvity data

for heptane. Calculations for numerous other fuels have been made by Sealand

Corp. (using a slightly less general model) and are reported elsewhere [1].

Calculations were made for various locations of a single suction pipe,

for several air flow rates and several stable stratification temperatures.

The effect of varying the height of the suction above the tank top is shown in

figure 15 for a suction on the centerline of an idealized ship. Raising the

suction from .05 m above the tank top to 1.5 meters is seen to decrease the

evaporation rate by a factor of 3-1/4 for a stable stratification of 0.1°C

over the hold height of 20 m. Since the evaporation rate varies as flow to

the third power, to remove liquid with the suction at 1.5 meters at the same

rate as with it at .05 m the flow would have to be somewhat more than 34 times

as great for the 1.5 meter height (3.25 cubed). For a 1°C stable stratification this effect of suction height is seen, from figure 17, to be very much

more pronounced.

Moving the suction laterally at constant height has comparatively little

effect. This is illustrated in table 2 which lists the dimensionless mass

pick-up for two suction locations, one on the centerline and the other near

one side of the ship. Figure 16 presents the same information graphically.

However, since this calculation fails to account for the interference to

transverse flow occasioned by the vertical framing of the bulkhead, these

results should be viewed with caution. Figure 17-a shows the calculated

- 63-
pressure at the open end* of a pair of slots, one on the hold centerline

(solid line), the other the laat alot outboard (dots). Pressures for two

suet Lon heights are shown, .05 m, figure 17-a and 0.6 m, figure 1 7—b • Note

that, except for a region of about 0.2 dimensionless units around the height

of the suction, the pressures are essentially the same for the two slot

locations at a given suction height. Since the slot flow Is determined

primarily by the gradient of the pressure at y*0, little difference in the

slot flows should be expected. This is reflected In the data presented in

table 2. Note also the effect on the pressures below the suction of raising

it, l.e., compare, in figures 17-a and 17-b, the pressure behavior below the

pressure "spike". In figure 17-b, the pressure gradient is small between the

tank top, y-0, and a dimensionless height of about 0.1 (dimensional height

0.2 m). The very small pressure gradient at y-0 results in little flow

adjacent to the tank top and little scavenging of the flammable vapors. This

accounts for the behavior shown in figure 15 of mass removal rate with suction

he Ight.

Streamlines corresponding to the two sets of paired pressure

distributions, fLgure 17a and 17b, are shown in figures 18-a,b and figures

19-a,b. In these, the transverse scale has been expanded about 100 times

relative to the longitudinal scale. The drawings should be very attenuated In

width to properly represent the geometry but, If so presented, would be

unreadable. The first pair, figure 18, show the streamlines for the boundary

layer flow at the bottom of the slots for the suction only .05 m above the

tank top. A thin layer of gas flows down the sides of the containers and

spreads out over the slot bottom—streamlines originating at the sides. It

turns and flows along the slot bottom toward the open end of the slot. For

- 64-
the slot closest to the suction the streamlines are strongly bunched in the

center of the slot. For the slot farthest away from the suction the behavior

is similar but the bunching of the streamlines near the centerline of the slot

is less pronounced. As the suction is raised there is quite a noticeable

change in the flow which becomes more marked as the suction is raised. This

change is seen by comparing figures 18-a and 19-a. With the suction raised,

not all the flow leaves the slot along its bottom surface. Some returns to

the container sides and flows up, turning toward the main void as it

approaches the height of the suction. By the same score, not all the flow

descends in the thermal boundary layer next to the container sides to the slot

bottom although it does toward the rear of the slot. Some of the flow near

the opening into the void turns as it approaches the height of the suction and

exits directly. This flow is shown schematically in figure 10. Note that

flow descending all the way to the tank top has actually been over cooled. In

the stably stratified situation pertaining, it must be wanned again in order

to rise to the height of the suction. To gain heat it must pass close to

either the walls of the main void or of the slot. Obviously, for the suction

raised well above the tank top, some of the air finds it easier to seek the

slot side wall rather than the main void walls. This flow behavior was found

through the numerical calculation. Although it is completely plausLble , it

was not anticipated. As the suction is further raised the flow along the slot

bottom becomes still weaker and the tendency to return to the wall

decreases. It has almost completely dissappeared with the suction at 1 m,

figures 20a, b, and c.

The effect of increasing the stable thermal stratification is seen by

comparing figures 17-a and 21. Note the different ordinate scales of the

- 65-
pressure plots. In fLgures 17-a the temperature difference over the 20 m hold

height is 0.1°C while it is 1.0°C in figure 21. The dimensional suction

heights are the same but, due to the greater thermal stratification, the

dimensionless height at AT = 1.0°C is greater. In general, the effect of

increasing AT is to compress the vertical scale of the flow. With a higher

AT the same vertical suction height appears to the flow as further from the

tank top.

Increasing the stable temperature stratification acentuates the flow

effects as shown in figures 18, 19, and 20. The flow in the slot closest to

the suction bunches strongly but then spreads rapidly as the mouth of the slot

is approached, figure 21-b. Even with this low suction location some flow,

driven by the stronger stratification, returns to the wall. The flow in the

slot furthest from the suction is less strongly affected, figure 21-c.

The main computed results for a given hold configuration and temperature

stratification are expressed in dimensionless form for a nominal temperature

of 10°C. To obtain dimensional output the ventilation flow and spill liquid

partial pressure and diffusivity are needed. Some vapor pressure data as a

function of temperature can be found in the literature [2] and, typically, the

log of the vapor pressure is nearly linear with 1/T as shown for heptane in

figure 2. Although the theory for vapor pressure is very well developed [15],

a much simpler semi-empirical approach, Antoine's formula, is found in Hirata,

Oke and Nagahame [14]

log p = A = B/(C + T)

- 66-
where p is the part Lai pressure, T temperature and A, B, C constants unique to

the particular chemical.' Further, since Hirata et al give values for these

constants for a large number of chemicals their approach was used. DLffusLvity data was corrected for temperature using the empirical relatLon gLven in

[16] •

It Ls clear from figure 2 that the equilibrium vapor concentration is a

strong function of temperature. The numerical calculation obtains the

concentration for the input value of slot bottom temperature. However, the

flow, which depends strongly on the differential stratification temperature,

but only weakly on average ambient temperature, is computed using a nominal

ambient.

10. EXPERIMENTAL PROGRAM

An approximately 1/12 scale model of a much simplified containership hold

was built. It consisted of a box with inside dimensions 1.22 m long x 2.66 m

wide x 1.68 m high built of 0.65 cm fir plywood and 0.6 mm galvanized steel.

To simulate the cooling effect of seawater on the hull, the bottom and lower

74 cm of the two ends were steel, maintained at constant temperature by

circulating temperature controlled water through copper tubes soldered to the

steel. The remainder of the apparatus was of wood, and allowed to seek room

temperature. The apparatus was in a large temperature controlled room where

typically the temperature varied between 18 and 25°C with an average value of

23°C during the course of a day. Inside this "hold" were three plywood boxes

1.51 m high x 1.02 m long. The center box was 0.81 m wide and the two outer

boxes 0.61 m wide. These simulated stacks of containers. These three boxes

-67 -
were positioned to give two slots 4 cm wide between them and 2 cm wide slots

between them and the sides of the hold. All three were centered between the

long sides of the hold leaving two transverse voids 10 cm wide (fore and

aft). All the plywood surfaces were coated with epoxy base paint to prevent

take-up of the spill vapors. A simplified cross section of the apparatus Ls

shown in figure 22.

The flammable spill was represented by a shallow pan filled with an ethyl

alcohol-water mixture. The pan was 1.16 x 2.39 x 0.025 m and held about 40

liters of mixture. The pan was placed in the "hold" and the simulated

container stacks set in on pads 0.031 m high to allow the liquid to pass

freely under them. The choice of a water alcohol mixture as the spill liquid

was unfortunate. It was made for safety reasons. The flammable gas detector

to be used could detect the alcohol vapors at concentrations well below their

flammable limit and mixtures and temperatures were chosen so that only

nonflammable mixtures would exist. The mixtures themselves were too dilute to

support combustion. However, the concentration of the mixture was difficult

to control and changed substantially during the course of a test. Thus, there

was constant uncertainty about the equilibrium vapor pressure of the spill

liquid.

Reducing the physical scale requires that the initial temperature

difference be increased if similar flow conditions are to prevail. The

3 2 dimensionless Grashof Number, GR = (gd /v ') (AT/To) (d/h)Pr, must be kept

constant. WLth a 1/12 reduction in physical size the ratio d/h was increased

but, in the model, the temperature difference top to bottom for similar stable

stratification should be increased by a factor of 90. In practice a tempera-

- 68-
ture difference of about 15°C could be achieved, equivalent to a full-scale

temperature difference, AT, of only 15/90 = 0.17°C. Calculations Ln the

previous section were for AT's of 0.1 and 1.0°C. The stable situations tested

are for weak stratification compared to full scale. For the unstable case,

model temperature difference of only 5°C was achieved. Thus the full scale

instability simulated was quite weak.

Due to the limited budget and time, no air flow measurements inside the

"hold" were attempted so it was never established if the actual flows

resembled those predicted by the calculations nor if there were significant

circulating flows, as might arise from a transverse temperature gradient.

Some tests were run in the configuration shown in figure 22 and some with the

inlet and exhaust connections reversed. For all tests the cooling water was

turned on and adjusted to the test conditions the day before the spill liquid

was to be loaded and data taken.

Results for a stably stratified condition are shown in figure 23 for

several air change rates. Note, however that, again a scaling factor enters,

model air changes per hour are not the same as full-scale. The behavior

during the first 15-30 minutes is erratic because of the transients associated

with loading the liquid in the pan, achieving steady liquid temperature and

ventilation conditions. The qualitative trend after 30 minutes, for both

suction locations, is to reduce the combustible vapor concentration in the

exhaust as air flow is increased. With the exhaust at the bottom, significantly more vapor is removed than with it at the top for the same air change

rate.

- 69-
When the behavLor with a stable temperature gradient Ls compared to that

wLth an unstable gradLent, it is necessary to recall that, in thLs apparatus,

the upper portion was always nominaly at room temperature. When a stable

temperature gradient was called for, the lower portion and the liquid pan were

cooled; when an unstable gradient was desired they were warmed. Thus, with

the same spill liquid, the equilibrium vapor concentration in the stable case

(cool liquid) was considerably less than for the unstable case (warm

liquid). At zero air changes per hour a survey of vapor concentration versus

height showed little variation. At 1 air change per hour with a bottom

exhaust for the unstable case the vapor concentration, except very close to

the surface of the pan, was quite uniform and about 40% of its equilibrium

value. For the stably stratified case the vapor concentration decreased near

the pan more slowly with height than in the unstable case but leveled out at

about 30% of its equilibrium value about 10 cm above the bottom and decreased

slowly with height above that, figure 24. Shifting the exhaust from the

center to one end had no effect to the accuracy of these tests. The behavior

in the stably stratified case suggest that a significant secondary mixing flow

was present, but that a mild degree of stability was achieved.

Overall the experimental results were ambiguous to mildly encouraging.

They also illustrate the inadvisability of conducting convection experiments

hastily and with a tight budget. The experimental apparatus was built by

Mr. W. Bailey and the data taken by Mr. S. Steel, both of NBS.
11. CONCLUSIONS

One of the more interesting aspects of thLs study of a significant hazard

to shipping was the almost total lack of previous scientific research at the

time the initial decision of the International Maritime Consultative Organization relative to containership ventilation was made. Our limited study

revealed many unanswered questions about fluid flow and heat and mass transfer

in the context of maritime safety. It shed some light on one particular

situation — the ventilation of a stably stratified containership hold. Many

assumptions had to be made — when a surface temperature was needed, we

assumed it was known, when geometric simplifications had to be made, it was

assumed that associated complicated flow phenomena were not present. Despite

these simplifications and idealizations, the work reported here is, to the

best of our knowledge, the only analysis in existence that attempts to study

the efficiency of ventilation in a realistic hold geometry and thermal

environment.

Although much more detailed research should be done on a variety of

specific points, we believe that some important conclusions can be drawn from

our study and that these conclusions will stand the test of time.

(1) In a stably stratified containership hold, almost all the

extracting capability of a ventilation system will be

concentrated at the vertical level of the suction.

(2) In order to extract any significant amount of spilled

material, it is essential to place the suction as close to

the hold bottom as is feasible.

- 71 -
(3)

With few exceptions, the natural tendency of vapors from

spLlled liquids will be to concentrate near the hold

bottom. Therefore it Is not efficient to try to design the

ventilation system so as to mix the gas throughout the hold. (4)

Ventilation expressed simply as air changes per hour (for

some arbitrary hold loading condition — empty or full) is a

poor measure of performance. The performance will be

sensitive to the thermal environment, degree of stratification and spacing between containers.

(5)

The lack of the ability to analyze the thermal environment

of a containership hold is a major impediment to systematic

study of hold ventilation. There is important need for

research in this area.

12. REFERENCES

[1] von Iperen, Willem, H.P., A Study of Ventilation of Containership Holds

for Carriage of Flammable Liquids, unpublished, June 18, 1979, Sea-Land

Service, Inc. P.0. Box 1050, Elizabeth, NJ 07207.

[2] Hodgman, C.D., Handbook of Chemistry and Physics, 27th Edition, Chemical

Rubber Publishing Co. , Cleveland, OH, 1943.

[3] Lewis, B. and von Elbe, G. , Combustion, Flames and Explosion of Gases,

Academic Press, New York, 1961.

[4] Turner, J.S., Buoyancy Effects in Fluids, Cambridge University Press,

Cambridge, MA, 1973.

[5] Birikh, R.V., Gershuni, G.Z., Zhukhovitskil, E.M. and Rudakov, R.N.,

Stability of the Steady Convective Motion of a Fluid with a Longitudinal

Temperature Gradient, Journal of Applied Mathematics and Mechanics,

Vol. 33, No. 6, 1969, pp. 937-947, Pergammon Press, Elmsford, NY

(translated from the Russian version).

-72-
[6] Schllchting, H. , Boundary Layer Theory, McGraw-Hill, New York, 1955.

[7] Batchelor, G.K., An Introduction to FluLd DynamLcs, Cambridge UnLversLty

Press, Cambridge, MA, 1967.

[8] Prandtl, L. , Essentials of Fluid Dynamics, Blackie, London, 1952.

[9] Gill, A.E., The Boundary Layer ‘Regime for Convection in a Rectangular

Cavity, J. Fluid Mech. , 26, p. 513, 1966.

[10] Elder, J. W. , Laminar Free Convection in a Vertical Slot, J. Fluid Mech.

23, p. 77, 1965.

[11] Carrier, G.F., Analytical Approximation Techniques in Applied Mathematics, J. Soc., Indust. Appl. Math., 13, p. 68, 1965.

[12] Lighthill, M.J., Contributions to the Theory of Heat Transfer Through a Laminar Boundary Layer, Proc. Roy. Soc. A, 202, p. 359, 1950.

[13] Abramowitz, M. and Stegun, I. A. , Handbook of Mathematical Functions,

National Bureau of Standards AMS 55, U.S. Govt. Printing Off Lee,

Washington, DC, 1964.

[14] Hirata, H. , Oke, S. and Nagahame, K. , Computer Aided Data Book of VaporLiquid Equilibrium, Kodanska Limited, Elsevier Scientific Publishing

Co., Tokyo/New York, 1975.

[15] Summerf ield. A., Thermodynamics and Statistical Mechanics, Academic

Press, New York, 1956.

[16] Eckart, H.R.G. , Introduction to Heat and Mass Transfer, McGraw-Hill, New

York, 1963.

I -73-
X Vi

4-1 0) -4 c

-4

u co oc X 4J c V e —

c o e o to a Vi —i OJ *4-4 zz JZ X 4-1 ~o u to V c a) c 3 JZ

—1 o ZJ

to JZ r-4

4-1

X 3 c JO o X o —i <u JZ

r—i 4-1 4-1 to JZ o oo 4-1 —4 -4 4-4 a. Vi <0 5^ rH

4-1 (V vr a> 3 4-1 J3

4-1

>4-1

O

OH o

4-1 co • a —( C c

4-1 o <0 o —

V 4-1 00 X (J c

<u —4

x 00 X o o •a to w —

4-1 o e X

OJ Vl3

CO ~4

lu I •3C

Ik
PRESSURE

(mm

Hg)

Figure 2 Vapor pressure versus temperature for heptane.

Rich and lean limits are for an air environment

75
Figure

3.

Idealized

containership

hold

loaded

with

containers
A

h

i

Figure 4. Geometry of the space between two stacks of containers

(inter-container slot)

77
=PL

(n,y) [

0Ar=

sin

(7

t£)

K

03

00 \ iX

II *0L Us*.

\oo "ro o

GO l Cl

d

78

FLgure

5.

Conformal

mappings

used

to

obtain

the

Greens

function,

G,

In

equation

24
secondary

slot

exit n

VI

VI

CM

79

Figure

6.

Plan

view

to

the

end

void

showing

the

locatLon

of

the

forced

ventilation

extraction

point

(suction).

Ventilation

flow

enters

from

the

top

of

the

void

and

from

the

interconta

Iner

slots

(one

ofN

slots

indicated

by

a

vertical

line)
80

Figure

7.

Cross

section

of

the

bottom

of

an

intercontainer

Velocity

distribution

of

the

gas

as

it

approaches

slot

bottom

is

indicated

next

to

each

wall
U

(y,z)

versus z

for

fixed y

Figure

8.

Universal

velocity

profiles

for

the

axial

flow

in

the

boundary

layer

of

wan

a

slot

bottom.

The

flow

is

symmetrical

about

the

slot

centerline

so

only

one

half

the

slot

width

is

shown.

The

velocity

is

zero

at

the

slot

bottom

and

increases

toward

an

asymptotic

profile

as

height,

y,

above

the

bottom

increases.
W(y,z) =I

mu\ K

<=>

CM

io cd

m

CM CMm " CD

ca i e CD —4 4J i—4 a >4-1 32 •H 3 CO rH 4J >43 o CO o 1—

CO 32 4J V4 *M •H

4)

COa

U 3 C o o

CO o o Vi —i co Vi rH a) 4-1 <D V >4-1 >, N o CO > rH CO

CO 0) c 00

a CD

C £ o •H E u

cO >. o

Vi o >i co c u CO 4J c0 —I • -H

<d s CDu

C E £ o e o CO O

4-1 4-1 •H 1—

4-4 4-1 rH a> •a 4-1 Vi o V4 > Vi O o 33 <D c0 33

(4-1 4-1 0) 3

4-1 c rC

O 4-1 CO o a) H 4-1 o

ai rH o rH i—

00 CO CO 4J • 11

<4-1 CO

OC 00 CD o i—

3 co 32

Vi >4-1 COO a> 43

a o 32 Vi V CO O CD >* Vi 32 C >

4-1 a> 4-1 CO -3

O

-3 * —1 33 o C0 4-1 TJ co o rH 3 32 C

rH

O 4J cO

—

<u >> 33a > Vi C0 •H E

(0 3 O •> 1—

TJ

a 4J 43

CO c •H •u 4-1 32

CO 3 Vi

O

O oc

Vi

O 4-1 rH 33

u

a> 33 at CO CD > E 4J 32 •H a) E CD O C 32 R 32 rH CO 33 4-1 CO 4-1 (0 CO

CT>

<U

Vi3

00

—i

Ph o

82 . |
Figure 10. Perspective sketch of an intercontainer slot showing

air flow streamlines. The air picks up evaporated

spill vapor as it moves in the boundary layer along

the bottom of the slot
Figure 11 Simplified flow diagram for the computer program

84
lO

co

(o'Ajan

Figure

12.

Universal

slot

bottom

axial

velocity

plotted

versus

heLght

for

a

fixed

transverse

location

(centerline)
STREAM

FUNCTION

Figure 13 S bottom stream functLon near the Intersection

of :he slot with the hold end void as a function of

d' ance from the Intersection (at x = -1/?)

86
Y

(dimensionless

height)

Figure 14 . Dimens Lonless slot pressure as a function of

dimensionless height above the slot bottom for two

slots: solid line, slot nearest the suet Lon and

dotted line, slot furthest from the suction.

Pressures calculated single precision. Compare wLth

figure 20-a which is the same case calculated Ln double-precision 87

;

i
>>

X) 4-1 14-1 J2 0

.H SO

co -1 03 > 34 4-4 o SI c 5 3

n> c O u o

—4

-C 4-4 c O 4-4 (3 3 c

03 34

C Pi O <4-4 34 —1 o 4-4

4-4 14-1 <0 C —4

V4 o T3 O -r-4 C. 4-> O

CO

O 3 > C 4-4 04 3

14-4 t-l >> o

-Q to 14—4 Cu 03 6 e 3 to O o | 4-4 1—

s* c 4-4 4-1 o o o CO —* -4 X3 o

Q. 4-4 CO 4-4 44-4 03 .-I o •?-< 03 —4 i—4 4-4 CO 4-4 03 CO e c Pi <u 34 4-4 "O > X GO

04 4-1 4-4 -Q 34 3 0) 34 r—4 O- CJ > x> e Pi C CO O O X 4-1 U U-4 CO 03 m

34u3

60

T-l Pu

HEIGHT |m)

88

J
U OQ O Jrf

i-H CJ C

CO CO O

4J —

<

4-1 X 4-1 o o O 3

G •—i CO O ^ -* G u in o

o u x Cow

3 i-H CO 4H CO I-H

CO i—I 4-1 -H G

co <u

cO oO >

G

Q. —l XJ 3 C <U

I -H O

X co W

O 4_l O

•H C 4H

a. o

cj o

co ^

10 D H

•« <u

<y X

C w

SowX o

CO 4H

CO CO

<D H 4 CD Cow

O 4-1 <y -H C

CO U —

*

C 0) CO O X W

e § c

-h 3 O

a c o

O XJ X W

CO 3 o ox

C w

O 3

•H o

4-1 o <u 3 X

CO 4J

a> co 5 G

i—

I

o

—I

•to 4-1 •r( O

rH 3 o co CO

a>

•• c

CO X

G x

I

o

•H X)

4-1 0)

CO w X)

CJ 4J I-H o o o

rH X5 X

- X

CUu

3

OO

p4

89
Y

(dimensionless

height]

Figure 17. Dimensionless pressure as a function of dimensionless

height (double precision calculation) stable stratification 0.1°C over height of hold

P (dimensionless pressure)

Figure 17a. Physical height of suction 0.05 m, dimensionless

height 0.025

90
Y

(dimensionless

height)

)

I M L I I 0.3 0.5 0.7 0.9 1.1 1.3 1.5

P [dimensionless pressure)

Figure 17b. Physical height of suction 0.60 ra, dimensionless

height 0.294

91
X

(dimensionless

length)

FLgure 18. Slot bottom boundary layer streamlines corresponding

to i.he pressures shown in f Lgure 17a

Z (dimensionless width)

FLgure 18a. Streamlines for slot at center (nearest suction)

92
x

[dimensionless

lengxnj

Figure 18b. Streamlines for slot at hold sLde (furthest from

suction)

93
X

(dimensionless

length]

Figure 19. Slot bottom boundary layer steamlines corresponding

to the pressures shown In f Lgure 17b

94
X

(dimensionless

length)

Figure 19b. Streamlines for slot at hold side (furthest from

suction)

95
Y

(dimensionless

height)

P [dimensionless pressure)

Figure 20a. Dim?:;.sionless pressure as a function of dimensionless

height (double precision calculation, compare with

fLgvre 14). Suction 1 m above hold bottom

96
X

(dimensionless

length)

Z (dimensionless width)

Figure 20b. Slot bottom boundary layer streamlines cor sspondLng

to the pressure shown in figure 20a. Centerline

slot (nearest suction)

97
X

(dimensionless

length)

Figure 20c. Slot bottom boundary layer streamlines corresponding to the

pressures shown in figure 20a. Side slot (furthest from

suction)

98
99

Figure

21a.

Dimensionless

slot

pressure

as

a

function

of

dimensionless

height

(double

precision-calculation)

stable

stratification

1°C

over

height

of

hold

(compare

with

figure

17)
X

(dimensionless

length)

-1.75 -1.25 - 0.75 - 0.25 0.25 0.75 1.25

Z (dimensionless width)

Figure 21b. Slot bottom boundary layer streamlines corresponding

to the pressures shown in figure 21a. Centerline

slot (nearest suction)

100
X

[dimensionless

length)

Z [dimensionless width)

Figure 21c. Slot bottom boundary layer streamlines corresponding

to the pressures shown in figure 21a. Side slot

(furthest from suction)

101
CONTAINER

STACK

Figure

22.

Simplified

schematic

of

experimental

apparatus
a?

"D

oo

UJ

CD

00

103

Figure

23.

Concentration

of

combustible

vapor

in

the

exhaust

duct

for

several

ventilation

rates

and

two

suction
CONCENTRATION

VERSUS

HEIGHT

AT

1

AIR

CHANGE/HR

NORMALIZED

BY

SATURATED

VAPOR

CONCENTRATION

N0I1VM1N33N0Q MOU JO 30VH3AV/ NOIlVdlNJONOO

Figure 24. Concentration to combustible vapor inside the

experimental enclosures versus height of the sampling

point. One air change per hour forced ventilation

flow, low suction

104

:

HEIGHT,

INCHES
Table 1. Vertical pressure gradient effectiveness f ( on) dependence upon

stratification parameter u> defined in Eq. (10) of text. f (w)

is the ratio of vertical to horizontal flow capable of being

produced by a given pressure gradient.

0) f(<d)

0

12

3

4

5

1.3299

.0878

.0276

.0117

.0060

105
Table 2. Dimensionless Mass Pick-Up

Suet Lon Height 5 cm Mass PLck-Up

Slot # Centerline Suction Side Suet Lon

1 2.01 2.72

2 2.05 2.49

3 2.12 2.29

4 2.25 2.16

5 2.37 2.00

6 2.51 1.91

7 2.34 1.82

8 2.18 1.74

9 2.12 1.74

10 2.03 1.70

11 1.95 1.66

106
APPENDIX 1.

Example of Input Data

107
TEST

CfiSE

5/

rEPTANE

SPILL

(C.P.)

0.1

ICO.

4497.13

96.15

279. C9
APPENDIX 2.

Output for the Input Data of Appendix 1

109

=Cv=T5r

INC£P?*. 3

ENT

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NBS-114A (rev. 2 -sc) U.S. DEPT. OF COMM. 3. Publication Date

May 1983

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 83-2665

2. Performing Organ. Report No

4.

TITLE AND SUBTITLE

An Investigation of the Forced Ventilation

in Containership Holds

5.

AUTHOR(S)

Howard R. Baum and John A. Rockett

6.

PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8 . Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

U.S. Coast Guard

400 Seventh Street, SW.

Washington, D.C. 20590

10.

SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

hi bliography or literature survey, mention it here)

An analysis of the fluid flow and mass transfer induced by ventilation systems

in containership holds was carried out. The work was performed in support of

the U.S. position before a committee of the International Convention on Safety

to Life at Sea. The analysis consists of a detailed calculation of the forced

motion through an interconnected set of narrow, stably stratified vertical air

passages which represent an idealized containership holds the results of this

calculation are then used in a study of the concentration boundary layers

formed by the pickup of spill material assumed to lie at the bottoms of the air

passages. The result is a set of formulae which determine the rate of extraction of spill material as a function of hold geometry, ventilation parameters,

and ambient stratification. The results are incorporated in a computer program

which is described in detail. A variety of computed results are presented,

together with a listing of the program. The results indicate the crucial

importance of locating the extractor as close to the hold bottom as technically

possible.

12.

KEY WORDS (Six to twelve entries ; alphabetical order; capitalize only proper names; and separate key words by semicolon s) Cargo ships; fires hazardous materials; modeling; stratified flow;

ventilation

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

[j£] Unlimited

[~~| For Official Distribution. Do Not Release to NTIS

[~] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

175

15. Price & Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $16 .00

USCOMM-DC 6043-P80

MM

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